A PROBABILITY BASED FAILURE MODEL FOR
COMPONENTS FABRICATED FROM ANISOTROPIC
GRAPHTIE
CHENGFENG XIAO
Bachelor of Science
Sun Yat-Seng University, Guangzhou, China
Master of Mechanical Engineering
Sun Yat-Seng University, Guangzhou, China
Submitted in partial fulfillment of the requirements for the degree
DOCTOR OF PHILOSOPHY IN CIVIL ENGINEERING
At the
CLEVELAND STATE UNIVERSITY
i
April 2014
ii
ACKNOWLEDGMENTS
iii
A PROBABILITY BASED FAILURE MODEL FOR
COMPONENTS FABRACATED FROM ANISOTROPIC
GRAPHTIE
CHENGFENG XIAO
ABSTRACT
The life prediction of graphite components in the researching Generation IV
nuclear power plants under complex loads has been studied in the past decade. The stress
tests of graphite H-451 samples show different strength in tension and in compression
and slight anisotropic material property. An appropriate failure criterion model
describing material behavior appropriately and an available reliability assessment
technique have been explored.
This work provides review of property of a graphite material and failure criterion
models applied widely. The failure criterions models selected, from simple Von Mises
failure criterion model with one parameter to complex Green-Mkrtichian failure criterion
iii
model with three parameters, are recorded. To accomplish the convenient formulation
the invariant theory and Cauchy-Hamilton theory are utilized. Compared with other
failure functions the quadratic form of the original isotropic Green-Mkrtichian failure
function is relatively easily extended to an anisotropic formulation. The new integrity
base is created through using a material unit vector. The new model has six strength
parameters, including four uniaxial strength and two multiaxial strength ones.
Comparison with the Burchell’s data shows that the anisotropic formulation of Green-
Mkrtichian failure criterion describes the material behavior of graphite successfully.
The work also transforms the deterministic failure model to a probabilistic failure
model through considering the strength variables as random variables. The graphite is
characterized by Weibull distribution. The direct integration of the reliability function is
tough to obtain a closed form with the complex anisotropic Green-Mkrtichian failure
function. For this reason the reliability index is introduced to calculate the failure
probability. In order to obtain the First Order Reliability Method (FORM) is
considered here. Some optimization methods are utilized to determine the most probable
point (MPP) that corresponds to . The Rackwitz-Fiessler method transforms Weibull-
variables to normal-variables. And the failure function expands with the first order
Taylor set. The failure probability of unit graphite volume is easily calculated with the
reliability index . The Monte Carlo simulation with Importance Sampling is used to
increase the accuracy of the failure probability results by . The simulation is working in
iv
a vanity area around MPP. The important samples are created with the sampling density
function selected. All of the programs run in MATLAB environment. The uniaxial
results are compared with the theoretical line by the Weibull reliability formulation. And
the different curves with three failure probability, 5%, 50% and 95, are shown on the
stress plane can compared. Importance conclusions are obtained.
v
TABLE OF CONTENTS
A PROBABILITY BASED FAILURE MODEL FOR COMPONENTS FABRICATED
FROM ANISOTROPIC GRAPHTIE...................................................................................i
ACKNOWLEDGMENTS...................................................................................................ii
ABSTRACT.......................................................................................................................iii
CHAPTER I GRAPHITE COMPONENTS IN NUCLEAR REACTORS.........................3
CHAPTER II STRENGTH BASED FAILURE DATA.....................................................8
2.1 Integrity Basis..................................................................................................9
2.2 Invariants of the Cauchy and Deviatoric Stress Tensors...............................12
2.3 Graphite Failure Data....................................................................................17
2.4 One-Parameter Model: the von Mises Failure Criterion...............................17
CHAPTER III TWO AND THREE PARAMETER FAILURE CRITERIA....................27
3.1 Two-Parameter Model: the Drucker-Prager Failure Criterion......................27
3.2 Three-Parameter Model: Willam-Warnke Failure Criterion.........................43
CHAPTER IV ISOTROPIC GREEN-MKRTICHIAN.....................................................56
4.1 Integrity Basis and General Functional Form................................................57
1
4.2 Functions and Associated Gradients by Stress Region..................................58
4.3 Defining Relationships Between Functional Constants.................................66
4.4 Functional Constants in Terms of Strength Parameters.................................74
CHAPTER V ANISOTROPIC GREEN-MKRTICHIAN MODEL.................................83
5.1 Integrity Base for Anisotropy........................................................................84
5.2 Functions and Associated Gradients by Stress Region..................................85
5.3 Defining Relationships Between Functional Constants.................................92
5.4 Functional Constants in Terms of Strength Parameters...............................116
CHAPTER VI MONTE CARLO METHODS USING IMPORTANCE SAMPLING. .144
6.1 Monte Carlo Simulation - Concept..............................................................147
6.2 The Concept of Importance Sampling Simulation......................................149
6.3 Application to the Green-Mkrtichian Limit State Function.........................154
6.4 Uniaxial Strength and Reliability...................Error! Bookmark not defined.
6.5 Contours of Equal Reliability......................................................................164
CHAPTER VII SUMMARY AND SOLUTIONS..........................................................169
2
CHAPTER I
GRAPHITE COMPONENTS IN NUCLEAR REACTORS
As discussed by Saito (2010) nuclear energy plays an important role as a means to
secure a consistent and reliable source of electricity that can easily help utilities meet
system demand for the nation’s power grid and do so in a way that positively impacts
global warming issues. Proposed system designs for nuclear power plants, e.g., the
Generation IV Very High Temperature Reactors (VHTR) among others, will generate
sustainable, safe and reliable energy. The nuclear moderator and major structural
components for VHTRs are constructed from graphite. During operations the graphite
components are subjected to complex stress states arising from structural loads, thermal
gradients, neutron irradiation damage, and seismic events, any and/or all of which can
lead to failure. As discussed by Burchell, et. al. (2007) failure theories that predict
reliability of graphite components for a given stress state are important.
Graphite is often described as a brittle or quasi-brittle material. Tabeddor (1979)
and Vijayakumar, et al. (1987, 1990) emphasize the anisotropic effect the elongated grain
graphite structure has on the stress-strain relationship for graphite. These authors also
discuss the aspect that the material behaves differently in tension and in compression.
These two properties, i.e., material anisotropy and different behavior in tension and
compression, make formulating a failure model challenging.
3
Classical brittle material failure criteria can include phenomenological failure
criteria, as well as fracture mechanics based models. The approach taken in linear elastic
fracture mechanics involves estimating the amount of energy needed to grow a pre-
existing crack. The earliest fracture mechanics approach for unstable crack growth was
proposed by Griffiths (1920). Li (2001) points out that the strain energy release rate
approach has proven to be quite useful for metal alloys. However, linear elastic fracture
mechanics is difficult to apply to anisotropic materials with a microstructure that makes it
difficult to identify a “critical” flaw. An alternative approach can be found in the
numerous phenomenological failure criteria identified in the engineering literature.
Popular phenomenological failure criteria for brittle materials tend to build on the
one parameter Tresca model (1865), and the two parameters Mohr-Coulomb failure
criterion (1776) that has been utilized for cohesive-frictional solids. Included with these
fundamental model is the von Mises criterion (1913) (a one-parameter model) and the
two parameter Drucker-Prager failure criterion (1952) for pressure-dependent solids. In
the past these models have been used to capture failure due to ductile yielding. Paul
(1968) developed a generalized pyramidal criterion model which he proposed for use
with brittle material. In Paul’s work, an assumption that the yield criteria surface is
piecewise linear is utilized which is similar to Tresca’s model. The Willam and Warnke
(1975) model is a three-parameter model that captures different behavior in tension and
4
compression exhibited by concrete. Willam and Warnke’s model is composed of
piecewise continuous functions that maintain smooth transitions across the boundaries of
the functions. The proposed work here will focus extensively on models similar to
Willam and Warnke’s efforts.
With regards to phenomenological models that account for anisotropic behavior
the classic Tsai and Wu (1971) failure criterion is a seminal effort. Presented in the
context of invariant based stress tensors for fiber-reinforced composites, the Tsai-Wu
criterion is widely used in engineering for different types of anisotropic materials. In
addition, Boehler and Sawczuk (1977) as well as Boehler (1987, 1994) developed yield
criterion utilizing the framework of anisotropic invariant theory. Yield functions can
easily serve as the framework for failure models. Subsequent work by Nova and
Zaninetti (1990) developed an anisotropic failure criterion for materials with failure
behavior different in tension and compression. Theocaris (1991) proposed an elliptic
paraboloid failure criterion that accounts for different behavior in tension and
compression. An invariant formulation of a failure criterion for transversely isotropic
solids was proposed by Cazacu et al. (1998, 1999). Cazacu’s criterion reduces to the
Mises-Schleicher criterion (1926), which captured different behavior in tension and
compression for isotropic conditions. Green and Mkrtichian (1977) also proposed
5
functional forms account for different behavior in tension and compression. Their work
will be focused on later in this effort.
In addition to anisotropy and different behavior in tension and compression,
failure of components fabricated from graphite is also governed by the scatter in strength.
When material strength varies, it is desirable to be able to predict the probability of
failure for a component given a stress state. Weibull (1951) first introduced a method for
quantifying variability in failure strength and the size effect in brittle material. His
approach was based on the weakest link theory. The work by Batdorf and Crose (1974)
represented the first attempt at extending fracture mechanics to reliability analysis in a
consistent and rational manner. Work by Gyekenyesi (1986), Cooper, et al. (1986, 1988)
and Lamon (1990) are representative of the reliability design philosophy used in
analyzing structural components fabricated from monolithic ceramic. Duffy et al. (1986,
1989, 1990, 1991, 1992) presented an array of failure models to predict reliability of
ceramic components that have isotropic, transversely isotropic, or orthotropic material
symmetry. All of these models were based on developing an appropriate integrity basis
for each type of anisotropy.
Given the discussion above establishing a single form invariant probabilistic
based failure model for components fabricated from graphite is the motivation behind this
dissertation. The probabilistic failure model must reflect the material behavior of
6
graphite. Through the application of invariant theory and the Cayley-Hamilton theorem
as outlined in Spencer (1984), an integrity basis with a finite number of stress invariants
will be formulated that reflects the material behavior of graphite. The integrity basis,
when posed properly, spans the functional space for the failure model. A model based on
a linear combination of stress invariants will capture the material behavior in a
mathematically convenient manner. This effort begins by proposing a deterministic
failure criterion that reflects relevant material behavior. When the model parameters are
treated as random variables, a deterministic model can be transformed into a probabilistic
failure model. Monte Carlo simulation with importance sampling will be used to
compute component failure probabilities. Throughout the dissertation the classical
models and the proposed failure criterion will be compared with the experiment results
obtained from Burchell (2007).
7
CHAPTER II
STRENGTH BASED FAILURE DATA
A function associated with a phenomenological failure criterion based on multi-
axial stress for isotropic materials will have the basic form
g = g (σ ij )(2.1)
This function is dependent on the Cauchy stress tensor, ij, which is a second order
tensor, and parameters associated with material strength. Given a change in reference
coordinates, e.g., a rotation of coordinate axes, the components of the stress tensor
change. The intent here is to formulate a scalar valued failure function such that it is not
affected when components of the stress tensors change under a simple orthogonal
transformation of coordinate axes. A convenient way of formulating a failure function to
accomplish this is utilizing the invariants of stress. The development below follows the
method outlined by Duffy (1987) and serves as a brief discussion on the invariants that
comprise an integrity basis.
8
2.1 Integrity Basis
Assume a scalar valued function exists that is dependent upon several second
order tensors, i.e.,
g = g ( A , B , C )(2.1.1)
Here the uppercase letters A, B and C are matrices representing second order tensor
quantities. One way of constructing an invariant formulation for this function is to
express g as a polynomial in all possible traces of the A, B and C, i.e.,
tr ( A ), tr ( A2 )
,tr ( A3 )
, … (2.1.2)
tr ( AB ), tr ( AC )
, tr (BC )
, tr ( A2 B )
… (2.1.3)
tr ( ABC ), tr ( A2 BC )
, tr ( A3 BC )
, … (2.1.4)
tr ( AB2C ),
tr ( AB3C ), … (2.1.5)
tr ( ABC 2 ),
)( 3ABCtr, … (2.1.6)
tr ( A2B2 C ),
)( 23 CBAtr, … (2.1.7)
where using index notation
tr( A ) = Aii
⋮(2.1.8)
9
tr( AB) = A ij B ji
⋮(2.1.9)
tr( ABC ) = A ij B jk Cki
⋮(2.1.10)
These are all scalar invariants of the second order tensors represented by the matrices A,
B and C. Construction of a polynomial in terms of all possible traces of the three second
order tensors is analogous to expanding the function in terms of an infinite Fourier series.
However a polynomial with an infinite number of terms is clearly intractable. On
the other hand if it is possible to express a number of the above traces in terms of any of
the remaining traces, then the former can be eliminated. Systematically culling the list of
all possible traces to an irreducible set leaves a finite number of scalar quantities
(invariants) that form what is known as an integrity basis. This set is conceptually similar
to the set of unit vectors that span Cartesian three spaces.
The approach to systematically eliminate members from the infinite list can best
be illustrated with a simple example. Consider
g = g ( A )(2.1.11)
By the Cayley-Hamilton theorem, the second order tensor A will satisfy its own
characteristic polynomial, i.e.,
A3 + k1 A2 + k2 A + k3[ I ] = [ 0 ](2.1.12)
10
where
k1 = −tr ( A )(2.1.13)
k 2 =( tr ( A ))2 − tr( A2)
2(2.1.14)
k 3 =−( tr ( A ))3 − (3) tr( A ) tr ( A2) + (2) tr( A3 )
6(2.1.15)
[ 0 ] = null tensor(2.1.16)
and
[ I ] = identity tensor(2.1.17)
Multiplying the characteristic polynomial equation by A gives
A4 + k1 A3 + k2 A2 + k3 A = [ 0 ](2.1.18)
Taking the trace of this last expression yields
tr ( A4 ) = − [ k1 tr ( A3) + k2 tr ( A2) + k3 tr ( A )](2.1.19)
and this shows that since k1, k2 and k3 are functions of tr(A), tr(A2), and tr(A3), then
tr ( A4 ) = g [tr ( A3) , tr ( A2 ) , tr ( A ) ](2.1.20)
11
Is a function of only these three invariants as well. Indeed repeated applications of the
preceding argument would demonstrate that tr(A5), tr(A6), … , can be written in terms of
a linear combination of the first three traces of A. Therefore, by induction
tr ( A p ) = g [tr ( A ) , tr ( A2 ) , tr ( A3) ](2.1.21)
for any
p > 3
Furthermore, any scalar function that is dependent on A can be formulated as a linear
combination of these three traces. That is if
g = g ( A )(2.1.22)
then the following polynomial form is possible
g = (k1 ) tr( A3 ) + ( k2 )tr ( A2 ) + (k3 ) tr ( A )(2.1.23)
and the expression for f is form invariant. The invariants tr(A3), tr(A2), tr(A) constitute
the integrity basis for the function f. In general the results hold for the dependence on
any number of tensors. If the second order tensor represented by A is the Cauchy stress
tensor, then this infers the first three invariants of the Cauchy stress tensor span the
functional space for scalar functions dependent onij.
12
2.2 Useful Invariants of the Cauchy and Deviatoric Stress Tensors
If one accepts the premise from the previous section for a single second order
tensor, and if this tensor is the Cauchy stress tensor ij, then
g(σ ij) = g ( I 1 , I 2 , I 3)(2.2.1)
where
I 1 = σ ii (2.2.2)
I 2 = (12 ) ( (σ ii )2 − σ jk σ kj )
(2.2.3)
and
I 3 = ( 16 ) [ (2 ) (σ ij σ jk σ ki ) − (3 ) (σ ii ) (σ jk σ kj ) + ( σ ii )3 ]
(2.2.4)
are the first three invariants of the Cauchy stress. Since the invariants are functions of
principle stresses
I 1 = σ1 + σ 2 + σ 3 (2.2.5)
I 2 = σ1 σ 2 + σ 2 σ3 + σ1 σ3 (2.2.6)
and
I 3 = σ1 σ 2 σ3 (2.2.7)
then
13
g (σ ij ) = g ( I 1 , I 2 , I 3)¿ g (σ1 , σ2 , σ3 )
(2.2.8)
Furthermore, the stress tensor ij can be decomposed into a hydrostatic stress component
and a deviatoric component in the following manner. Take
Sij = σ ij − ( 13 )σ kk δij
(2.2.9)
If we look for the eigenvalues for the second order deviatoric stress tensor (Sij) using the
following determinant
|Sij − Sδij| = 0(2.2.10)
then the resultant characteristic polynomial is
S3 − J 1S2 − J 2 S − J3 = 0(2.2.11)
The coefficients J1, J2 and J3 are the invariants of Sij and are defined as
J1 = S ii = 0(2.2.12)
J2 = (12 )S ij S ji
¿ (13 ) I12 − I2
(2.2.13)
and
14
N
1
2
3
O
d
),,( 321 P
rComponentDevatoric
ComponentcHydrostati
J3 = (13 ) Sij S jk Ski
¿ (227 ) I13 − (13 ) I 1 I 2 + I3
(2.2.14)
These deviatoric invariants will be utilized as needed in the discussions that follow.
Figure 2.3.1 Decomposition of stress in the Haigh-Westergaard (principal) stress space
2.3 Graphical Representation of Stress
In the Haigh-Westergaard stress space a given stress state (1, 2, 3) can be
graphically decomposed into hydrostatic and deviatoric components. This decomposition
is depicted graphically in Figure 2.3.1. Line d in figure 2.3.1 represents the hydrostatic
axis where 1 = 2 = 3 such that the line makes equal angles to the coordinate axes. We
define the planes normal to the hydrostatic stress line as deviatoric planes. As a special
15
case the deviatoric plane passing through the origin is called the plane, or the
principal deviatoric plane. Point P (1, 2 , 3) in this stress space represents an arbitrary
state of stress. The vector NP represents the deviatoric component of the arbitrary stress
state, and the vector ON represents the hydrostatic component. The unit vector e in the
direction of the hydrostatic stress line d is
e = 1√3
[ 1 1 1](2.2.15)
The length of ON, which is identified as , is
ξ = (OP ) e
= [σ1 σ2 σ3 ] 1√3
¿ [ 1¿ ] [ 1¿ ]¿¿
¿
¿
¿
(2.2.16)
The length of NP, which is identified as a radial distance (r) in a deviatoric plane, is
r = O P − O N
= [ σ1 σ2 σ3 ] − (I 1
√3 ) [1 1 1 ]
¿ [S1 S2 S3 ] (2.2.17)
From this we obtain
| r | = r2
¿ S12 + S
22 + S32
¿ 2J 2 (2.2.18)
16
such that
r = √2 J 2 (2.2.19)
One more relationship between invariants is presented. An angle, identified in the
literature as Lode’s angle, can be defined on the deviatoric plane. This angle is formed
from the projection of the 1 – axis onto a deviatoric plane and the radius vector in the
deviatoric plane, r . The magnitude of the angle is computed from the expression
θ = ( 13 ) cos−1[( 3√3
2 ) J3
(J 2)3 /2 ] (00 ≤ θ ≤ 600 )
(2.2.20)
As the reader will see this relationship has been used to develop failure criterion. It is
also used here to plot failure data.
We now have several graphical schemes to present functions that are defined by
various failure criterions. They are
a principle stress plane (e.g., the 1 - 2 plane);
the use of a deviatoric plane presented in the Haigh-Westergaard stress
space; or
meridians along failure surfaces presented in the Haigh-Westergaard stress
space that are projected onto a plane defined by the coordinate axes (
ξ−r ).
17
Each presentation method will be utilized in turn to highlight aspects of the failure
criterion discussed herein. We begin with one parameter phenomenological models and
then discuss progressively more complex models.
2.4 Graphite Failure Data
In the following section several common failure criterion models will be
introduced and the constants for the models are characterized using biaxial failure data
generated by Burchell (2007). For the simpler models Burchell’s (2007) data has more
information than is necessary. For some models all the constants cannot be approximated
because there is not enough appropriate data for that particular model. These issues are
identified for each failure model. Burchell’s specimens were fabricated from grade H-
451 graphite. There were nine load cases presented, including two uniaxial tensile load
paths along two different material directions (data suggests that the material is
anisotropic), one uniaxial compression load path, and six biaxial stress load paths. The
test data is summarized in Table 2.1. The mean values of the normal stress components
for each load path from Burchell’s (2007) data are presented in Table 2.2. In addition,
corresponding invariants are calculated and presented in Table 2.2 along with Lode’s
angle. All the load paths (#B-1 through #B-9) are identified in Figure 2.4.1.
18
2.5 The von Mises Failure Criterion (One Parameter)
The von Mises criterion (1913) is based on failure defined by the octahedral
shearing stress reaching a critical value. Failure occurs along octahedral planes and the
basic formulation for the criterion is
19
Table 2.1 Grade H-451 Graphite: Load Paths and Corresponding Failure Data
Data Set
Ratio
Failure Stresses
(MPa)
# B-1 1 : 0
10.97 0
9.90 0
9.08 0
9.22 0
12.19 0
11.51 0
# B-2 0 : 1
0 15.87
0 12.83
0 18.06
0 20.29
0 14.32
0 14.22
# B-3 0 : - 1
0 -47.55
0 -50.63
0 -59.72
0 -56.22
0 -48.19
0 -51.54
# B-4 1 : - 19.01 -8.94
7.68 -7.68
14.34 -14.16
8.93 -8.78
13.23 -13.14
9.21 -9.11
Data Set
Ratio
Failure Stresses
(MPa)
# B-5 2 : 1
7.81 3.57
8.54 3.89
11.2 5.6
13.00 6.42
11.54 5.76
12.12 6.03
# B-6 1 : 2
6.36 12.67
6.42 12.86
6.74 13.42
7.69 15.36
6.46 12.95
7.17 14.36
# B-7 1 : - 2
7.98 -15.99
5.50 -10.96
6.69 -13.37
10.49 -21.01
9.18 -18.30
11.31 -22.61
Data Set
Ratio
Failure Stresses
(MPa)
# B-8 1 : 1.5
6.69 10.03
6.51 9.78
8.07 12.11
9.13 13.74
6.11 9.19
9.24 13.91
9.93 14.93
8.93 13.41
7.20 10.79
# B-9 1 : - 5 6.35 -31.61
8.69 -43.44
7.40 -36.86
20
7.09 -35.30
5.94 -29.50
6.83 -32.83
8.06 -40.21
7.75 -38.58
21
)(11 MPa
)(22 MPa
#B-2 #B-6#B-8
#B-5
#B-1
#B-4
#B-7
#B-9
#B-3
Figure 2.4.1 Burchell’s (2007) Load Paths Plotted in a 1 – 2 Stress SpaceTable 2.2 Invariants of the Average Failure Strengths for All 9 Load Paths
Data Set (1)ave (MPa) (2)ave (MPa) (MPa) r (MPa)
# B-1 10.48 0 6.05 8.56 0.00o
# B-2 0 15.93 9.20 13.01 0.00 o
# B-3 0 -52.93 -30.56 43.22 60.00 o
# B-4 10.4 -10.3 0.06 14.64 29.84 o
# B-5 10.7 5.21 9.19 7.57 29.13 o
# B-6 6.81 13.6 11.78 9.62 30.05o
# B-7 8.53 -17.04 -4.91 18.41 40.88 o
# B-8 7.98 11.99 11.53 8.63 40.82 o
# B-9 7.26 -36.04 -16.62 32.79 50.99 o
22
1x
2x
3x
T
T
g (σ ij ) = g ( J2 )¿ AJ2 − 1¿ 0 (2.5.1)
To determine the constant A consider the following stress state at failure
σ ij = ¿ [ 0 0 0 ¿ ] [0 σ T 0 ¿ ] ¿¿
¿(2.5.2)
here is the tensile strength of the material, and for this uniaxial load case
J2 = ( 13 )σ
T2
(2.5.3)
Substitution of the value of the invariant J2 into the failure function expressed in (2.5.1)
yields
A = 3σ
T2(2.5.4)
So the failure function for von Mises criterion takes the form
g (σ ij ) = ( 3σ
T2 )J2 − 1
(2.5.5)
As mentioned previously we have several means to graphically present the von
Mises criterion. The von Mises failure function is a right circular cylinder in the Haigh-
23
1
1I2
3
Westergaard stress space shown as Figure 2.5.1. The axis of the cylinder is coincident
with the hydrostatic stress line. The right circular cylinder is open along the hydrostatic
stress line (i.e., no end caps) in either the tensile or compressive direction. Thus a
hydrostatic state of stress cannot lead to failure.
Figure 2.5.1 von Mises failure in Haigh-Westergaard stress space
Data set #B-2, which is tabulated in Table 2.3, represents a uniaxial tensile load
case. One can easily determine from this data that the mean strength is T = 15.93 MPa
and that
A = 3σ
T2= 3
(15.93 )2= 0 .0118
(2.5.6)
For a uniaxial load path where the stress is equal to the mean strength value for T , the
components of this stress state in the Haigh-Westergaard stress space are
24
r = 13. 01 MPaξ = 9 .20 MPa
(2.5.7)
Table 2.3 Invariants of the Failure Stresses for Load Path #B-2
11
(MPa)
22
(MPa)
(MPa) r (MPa)
15.87 0 9.16 12.96 0o
12.83 0 7.41 10.48 0o
18.06 0 10.43 14.75 0o
20.29 0 11.71 16.57 0o
14.32 0 8.27 11.69 0o
14.22 0 8.21 11.61 0o
The von Mises failure criterion is projected onto a deviatoric plane in Figure 2.5.2
utilizing these parameter values. The result of this projection is a circle. Figure 2.5.2
also depicts the data from load path #B-2 projected onto the deviatoric plane.
25
3
11
2
3
o0
o60o120
o180
o240o330
)01.13,20.9(
,MPaMPa
r
MPa5
MPa15
MPa20
MPa10
2
Figure 2.5.2 The von Mises criterion is projected onto a deviatoric plane (ξ=9. 20 MPa ) parallel to the deviatoric plan with T = 15.93 MPa
The von Mises failure criterion is also projected onto a 1 - 2 stress plane in
Figure 2.5.3. A right circular cylinder projected onto this plane presents as an ellipse.
An aspect of the von Mises failure model is that tensile and compressive failure strengths
are equal which is clearly evident in Figure 2.5.3. Obviously the Burchell (2007) data,
which is also depicted in Figure 2.5.3, strongly suggests that tensile strength is not equal
to the compressive strength for this graphite material.
26
)(11 MPa
)(22 MPa
)93.15,0(, 2211 MPa
Figure 2.5.3 The Von Mises criterion characterized with (T = 15.93 MPa) projected onto the 1 -2 principle stress plane depicting failure stress values for all load paths
The third type of graphic presentation is a projection of the von Mises failure
criterion onto the coordinate plane identified by the axes (- r). As noted above the von
Mises criterion is a right circular cylinder in the principal stress space. The function
depicted in Figure 2.5.4 results from a cutting plane that contains the hydrostatic line
coinciding with the axis of the right circular cylinder. The axis of the cylinder is
coincident with the – axis and all meridians will be parallel to the – axis. Thus all
meridians along the surface of the right circular cylinder representing the von Mises
27
)(MPar
)(MPa
0
1322
Jf
Tij
MPaMPar 01.13,02.9,
failure criterion are identical, i.e., the slope of all meridians is zero and the intercepts
along the r-axis are the same value. This is not the case for subsequent failure criterion
presented below.
Figure 2.5.4 The Von Mises criterion projected onto a meridian plane (T = 15.93 MPa)
Using the average normal strength values for the nine data sets from Burchell
(2007) nine sets of invariants are tabulated in Table 2.2. This information is used to plot
the average strength data Figure 2.5.4. As can be seen in the figure most of the averaged
Burchell (2007) data does not match well with the von Mises criterion characterized with
28
T = 15.93 MPa. The depiction in Figure 2.5.4 strongly suggests that , or I1, should be
considered in developing the failure function, i.e., something more than the J2 should be
used to construct the model. Since nuclear graphite is not fully dense, we will assume
that the hydrostatic component of the stress state contributes to failure. In addition, the
von Mises criterion does not allow different strength in tension and compression. When
other formulations are considered in the next chapter their dependence will have a well-
defined dependence on I1. This invariant will permit different strengths in tension and
compression, e.g., the classic the Drucker–Prager (1952) failure criterion outlined in the
next section.
As a final note on the one parameter models, the Tresca (1865) could have been
considered here. However, Tresca’s (1865) criterion, although based on the concept that
failure occurs when a maximum shear strength of a material is attained, is a piecewise
continuous failure criterion. Although later criterion considered here are similarly
piecewise continuous, the Tresca (1865) failure criterion does not mandate continuous
gradients at the boundaries of various regions of the stress space. This condition will be
imposed on the failure criterion considered later.
29
CHAPTER III
TWO AND THREE PARAMETER FAILURE CRITERIA
In the previous chapter Burchell’s (2007) failure data was presented in terms of a
familiar one parameter failure criterion, i.e., the von Mises criterion. The von Mises
failure criterion can be characterized through a single strength parameter – the shear
strength on the octahedral stress plane. In this chapter the view is expanded and details
of two and three parameter failure criterion are presented in terms of how well the
criterion perform relative to the mean strength of various load directions from Burchell’s
work (2007).
3.1 The Drucker-Prager Failure Criterion (Two Parameter)
In this section we consider an extension of the Von Mises criterion, i.e., a failure
model that includes the I1 invariant. This extension is the Drucker – Prager (1952)
criterion and is defined by the failure function
30
1x
2x
3x
T
T
g ( I 1 , J 2) = AI 1 + B √J2 − 1¿ 0 (3.1.1)
To determine the constants A and B first consider the following stress state at failure, i.e.,
a uniaxial tensile load
σ ij = ¿ [ 0 0 0 ¿ ] [0 σ T 0 ¿ ] ¿¿
¿(3.1.2)
here
I 1 = σT(3.1.3)
and
√J 2 = ( 1√3 )σT
(3.1.4)
Substitution of these invariants into the failure function (3.1.1) yields
A σ T + ( 1√3 ) BσT− 1 = 0
(3.1.5)
or
31
1x
2x
3x
C
C
A + ( 1√3 )B = 1
σT (3.1.6)
Next, consider the following stress state at failure under a uniaxial compression
load
σ ij=[0 0 00 −σ C 00 0 0 ]
(3.1.7)
where
I 1 = −σ C(3.1.8)
and
√J 2 = ( 1√3 )(σC )
(3.1.9)
Substitution of these invariants into the failure function (3.1.1) yields
A (−σC ) + ( 1√3 ) B σC − 1 = 0
(3.1.10)
or
32
−A + ( 1√3 ) B = 1
σ C (3.1.11)
Simultaneous solution of equations (3.1.6) and (3.1.11) yields
A = ( 12 ) ( 1
σ T−
1σC )
(3.1.12)
B = (√32 ) ( 1
σT+ 1
σC )(3.1.13)
Using the Burchell (2007) data from load path #B-2, the average tensile strength is
σ T = 15 . 93 MPa(3.1.14)
In a similar manner, using the load path #B-3, the average compressive strength is
σ C = 52. 93 MPa(3.1.15)
with these values of T and C the parameters A and B are
A = (12 ) (115 . 93− 1
52. 93 )= 0. 02194 MPa−1
(3.1.16)
and
B = (√32 ) (115 .93
+ 152.93 )
= 0.07073 MPa−1(3.1.17)
33
The Drucker-Prager (1952) failure criterion is projected onto the deviatoric plane
defined by
ξ = 9 . 20 MPa(3.1.18)
in Figure 3.1.1. There are an infinite number of deviatoric planes parallel to the -
plane. For the Drucker-Prager (1952) failure criterion each projection will represent a
circle with a different diameter on a different deviatoric plane. The graphical depiction
of the Drucker-Prager (1952) failure criterion on the deviatoric plane defined by
ξ = −30 .2 MPa(3.1.19)
also captures Burchell’s (2007) compressive data and is depicted in Figure 3.1.2. This
value of is obtained from averaging the compressive strength data along load path #B-
3. The invariants associated with the average strength from load path #B-3 are in Table
2.2. The invariants for all the failure strength data along load path #B-3 are presented in
Table 3.1. The Drucker-Prager (1952) failure criterion can be thought of as a right
circular cone with the tip of the cone located along the positive – axis. The cone opens
up along the – axis as becomes more and more negative. The negative value of
from equation 3.1.19 denotes a deviatoric plane beyond the - plane where = 0. The
failure criterion depicted in Figure 3.1.2 has a larger diameter than the failure criterion
depicted in Figure 3.1.1.
34
Figure 3.1.1 The Drucker-Prager (1952) criterion projected onto a deviatoric plane
(ξ=9. 20 Mpa) parallel to the -plane with T = 15.93 MPa, C =52.93 MPa
35
Figure 3.1.2 The Drucker-Prager (1952) criterion projected onto a deviatoric plane
(ξ=9. 20 Mpa) parallel to the -plane with T = 15.93 MPa, C = 52.93 MPa
Table 3.1 Invariants of the Failure Stresses for Load Path #B-3
11(MPa) 22(MPa) (MPa) r (MPa)
0 -47.55 -27.45 38.82 0o
0 -50.63 -29.23 41.34 0o
0 -59.72 -34.48 48.76 0o
0 -56.22 -32.46 45.90 0o
0 -48.19 -27.82 39.35 0o
0 -51.54 -29.76 42.08 0o
In Figure 3.1.3 the failure criterion is projected onto the 1 -2 stress plane and
compared with the entire data base from Burchell (2007). The right circular cone
typically projects as an elongated ellipse in this stress space. However, when the
Drucker-Prager (1952) failure criterion matches the mean failure stress along the 2 -
tensile load path (load path #B-2) and the 2 – compressive load path (load path #B-3) the
1 -2 stress plane slices through the right circular cone and produces a parabolic curve.
Note that the criterion does not match the Burchell (2007) data along the 2 tensile load
path (load path #B-1). The H-451 graphite Burchell (2007) tested is slightly anisotropic.
Moreover, the failure data from the biaxial stress load paths, #B-4 through #B-8 (the
36
exception is load path #B-9), are not captured at all since the failure curve is parabolic
and open along the equal biaxial compression load path. Anisotropic behavior and
parabolic failure curves (instead of elliptical curves) indicate a need for improvements
beyond the capabilities of the Drucker-Prager model in order to phenomenologically
capture the biaxial failure data.
37
)(11 MPa
)(22 MPa
)93.15,0(, 2211 MPa
)93.52,0(, 2211 MPa
Figure 3.1.3 The Drucker-Prager (1952) criterion projected onto the 1 -2 principle
stress plane (T = 15.93 MPa, C = 52.93 MPa)
The need for more flexibility is also evident when the Drucker-Prager (1952)
failure criterion is projected onto the stress space defined by the - r coordinate axes.
This projection is shown in Fig. 3.1.4 along with projections of the average strength
values from all nine load paths. As in the von Mises (1913) failure criterion, there is a
38
single meridian. The meridian for the Drucker-Prager (1952) failure criterion has a slope,
where the meridian for the von Mises (1913) failure criterion was parallel to the - axis.
As can be seen in Figure 3.1.4 three out of the nine average strength values align well
with the failure meridian projected into this figure based on the parameter values T =
15.93 MPa, and C = -52.93 MPa. These two parameters define the slope of the
meridian, and the meridian passes through the corresponding - r values, as it should.
The other six average strength values do not map closely to this single meridian for the
Drucker-Prager (1952) criterion. Keep in mind that the projection in Figure 3.1.4 is a
result of a cutting plane through the right circular cone and contains the hydrostatic stress
line. The data indicates that the failure function meridians should exhibit a dependence
on - defined by equation 2.2.20 and depicted in Figure 2.4.2. This can be accomplished
by including a dependence on the J3 invariant, and this is discussed in the next section.
39
)17.42,2.30(
,MPaMPa
r
)01.13,20.9(
,MPaMPa
r
)(MPar
)(MPa
Figure 3.1.4 The Drucker-Prager criterion projected onto the meridian plane(T = 15.93 MPa, C =52.93 MPa)
As noted above and depicted in Figure 3.1.3 the Drucker-Prager failure curve is
open along the equal biaxial compression load path. The following derivation will
demonstrate the transition from a parabolic (open) curve to an elliptic (closed) curve
based on the strength ratio C T. Consider the equal biaxial compression stress state
with BC < 0
σ ij = [−σBC 0 00 −σBC 00 0 0 ]
(3.1.20)
40
The corresponding deviatoric stress tensor is
Sij = [−σ BC
30 0
0 −σ BC
30
0 02σ BC
3]
(3.1.21)
The stress invariants of this state of stress are
I 1 = −2σ BC (3.1.22)
and
√J 2 = ( 1√3 )σ BC
(3.1.23)
Substitution of these invariants into the failure function (3.1.1) yields
A(−2σ BC) + B ( 1√3 ) σ BC− 1 = 0
(3.1.24)
or
σ BC = 1
−2 A + ( 1√3 ) B
(3.1.25)
Since BC > 0, then
σ BC = 1
−2 A + ( 1√3 ) B
> 0
(3.1.26)
41
which infers
2 A < ( 1√3 )B
(3.1.27)
This leads to
2( 12 ) ( 1
σ T− 1
σC ) < ( 1√3 )(√3
2 ) ( 1σT
+ 1σ C )
(3.1.28)
or
1σT
< 3σC
(3.1.29)
Thus
σC
σT< 3
(3.1.30)
When the ratio of compressive strength and tensile strength (C T) < 3, the Drucker-
Prager failure criterion projects an elliptical (closed) curve in the 1 -2 stress plane.
Consider the following biaxial state of stress
σ ij = [σ x 0 00 σ y 00 0 0 ]
(3.1.31)
The corresponding deviatoric stress tensor is
42
Sij = [2σ x−σ y
30 0
02 σ y−σ x
30
0 0 −σ x+σ y
3]
(3.1.33)
The stress invariants for this state of stress are
I 1 = σ x + σ y(3.1.32)
and
√J 2 = √( 13 )( σ
x2−σ x σ y+σy2)
(3.1.34)
Substitution of these invariants into equation (3.1.1) yields
f ( I 1 , J2 ) = A (σ x + σ y )
+ B√(13 ) (σ x2 − σ x σ y + σy2 ) − 1
¿ 0 (3.1.35)
or
√( 13 ) (σ x2 − σ x σ y + σ
y2 ) =1 − A (σ x + σ y )
B(3.1.36)
Squaring both sides yields
( B2−3 A )σx2 + (B2−3 A )σ y + (6 A−B2 )σ x σ
y2
+ 6 Aσ x + 6 Aσ y − 3 = 0 (3.1.37)
43
The shape of the failure criterion defined by equation (3.1.37) is determined by the values
of the two parameters A and B. Using Burchell’s (2007) tensile data where T = 15.93
MPa and a ratio of compressive strength to tensile strength of (C T) = 2, then C =-
31.86 MPa and the parameters A and B are
A = (12 ) (115 . 93− 1
31.86 )= 0. 01569 MPa−1
(3.1.38)
and
B = (√32 ) (115 . 93
+ 131 .86 )
= 0.08155 MPa−1(3.1.39 )
Equation (3.1.37) becomes
(0 . 0059 ) σx2 + ( 0. 0059 ) σ
y2 − (0 .0081 ) σ x σ y
+ (0 .0942 ) σx + (0. 0942 ) σ y = 3 (3.1.40 )
This expression is plotted in the 11 – 22 stress plane depicted in Figure 3.1.5
44
)(22 MPa
)(11 MPa
T 2,0, 2211
0,93.15, 2211 MPa
Figure 3.1.5 The Drucker-Prager (1952) criterion projected onto the 1 -2 principle
stress plane (T = 15.93 MPa, C = 31.86 MPa) and compared with the Burchell’s data
This combination of strength parameters leads to a biaxial strength of well over 900 MPa.
If Burchell’s (2007) compressive strength of C = 52.93 MPa is utilized from
along with a stress ratio (C T = 2), then the tensile strength is T =26.465 MPa. The
parameters A and B are
45
A = (12 ) (126 . 465− 1
52 . 93 )= 0. 009446 MPa−1
(3.1.41)
and
B = (√32 ) (126 . 465
+ 152 . 93 )
= 0.0490855 MPa−1(3.1.42)
Now
(0 .0021 ) σx2 + (0 . 0021 ) σ
y2 − (0.0029 ) σ x σ y
+ (0 .0567 ) σ x + (0 .0567 ) σ y = 3 (3.1.44)
This expression is plotted in the 11 – 22 stress plane depicted in Figure 3.1.6
NEED A THIRD FIGURE, 3.1.7 WHERE A NUMBER OF ELLIPSES ARE
SUPERIMPOSED IN THE SAME FIGURE. EACH ELLIPSE WOULD HAVE A
DIFFERENT STRENGTH RATIO OF C T RANGING FROM 1.0 TO 3.0 IN
INCREMENTS OF 0.25 OR 0.50.
46
47
MPa93.52,0, 2211
)(22 MPa
)(11 MPa
0,2/, 2211 C
Figure 3.1.6 The Drucker-Prager (1952) criterion projected onto the 1 -2 principle
stress plane (T = 18.25 MPa, C = -52.93 MPa) and compared with the Burchell’s data
Here the biaxial compressive strength is somewhat less than 1,100 MPa. In both figures,
i.e., Figure 3.1.5 and 3.1.6, closed ellipses are obtained which are important since all load
paths in this stress space eventually lead to failure. In Figure 3.1.3 the equal biaxial
compression load path was not bounded by the failure criterion given the strength
parameters from Burchell’s data. For all failure criterion considered, only those with
closed failure surfaces are relevant for consideration.
48
3.2 Willam-Warnke Failure Criterion (Three Parameter)
Willam and Warnke (1975) proposed a three-parameters failure criterion that
takes the shape of a pyramid with a triangular base in the Haigh-Westergaard (1 - 2 -
3) stress space. In a manner similar to the Drucker-Prager (1952) failure criterion, linear
meridians are assumed. However, the slopes of the meridians vary around the pyramidal
failure surface. The model is linear in stress through the use of I1 and √J 2 , which is
evident in the following expression
g ( I 1 , J 2 , J 3) = AI 1 + [ B(J 2 , J 3)] √J2 − 1¿ 0 (3.2.1)
Given the formulation above, in the Haigh-Westergaard stress space the Willam-
Warnke (1975) failure criterion is piecewise continuous with a threefold symmetry. This
symmetry is depicted in Figure 3.2.1 where the criterion is projected onto an arbitrary
deviatoric plane. The segment associated with 0o ≤ θ ≤ 60o
is presented. The
failure function is symmetric with respect to each tensile and compressive principle stress
axis projected onto the plane.
49
Figure 3.2.1 The Willam-Warnke (1975) criterion projected onto the deviatoric plane (
0o ≤ θ ≤ 60o)
As the deviatoric plane of the projection moves up the hydrostatic stress line in
the positive direction, the projection of the failure criterion shrinks. As the deviatoric
plane of projection moves down the hydrostatic line in the negative direction, the
projection of the failure criterion increases in size.
Willam and Warnke (1975) defined the parameter B from equation 3.2.1 in the
following manner
B = 1r (θ)
(3.2.2)
where r is a radial vector located in a plane parallel to the -plane. Willam and Warnke
(1975) assumed that when the failure surface was projected onto a deviatoric plane that a
50
1x
2x
3x
C
C
segment of this projection could be defined as a segment of an elliptic curve with the
following formulation
r (θ ) =2 rc(r
c2 − rt2)cosθ + r c(2 r t − rc )[ 4 ((r
c2 − rt2
)cos2 θ + 5 rt2 − 4 rt r c ]
1/2
4 ((rc2 − r
t2)cos2θ + (rc − 2 rt )2
(3.2.3)
Here is Lode’s angle, where once again
θ = ( 13 ) cos−1[( 3√3
2 ) (J 3)3
(J 2)3 ] (00 ≤ θ ≤ 600 )
(3.2.4)
When θ=0o, B=BT , r=rT and
rT = 1BT
(3.2.5)
Similarly, with θ=60o, B=BC , r=rC and
rC = 1BC
(3.2.6)
In order to determine the constants BT and BC consider the following stress state
σ ij = ¿ [ 0 0 0 ¿ ] [0 σ T 0 ¿ ] ¿¿
¿(3.2.7)
51
1x
2x
C
The deviatoric stress tensor is
Sij = ¿[− 13
0 0 ¿ ][ 0 23
0 ¿ ]¿¿
¿
(3.2.8)
and Lode’s angle as wells as the three invariants obtained are expressed as
( I1 , √J 2 , 3√J 3) = (σT , √33
σT ,3√23
σ T)(3.2.9)
θ = 00 (00 ≤ θ ≤ 600 )(3.2.10)
Substitution of the values of invariants into failure function given by equation (3.2.1)
yields
A( σT ) + (√33
BT )(σ T ) − 1 = 0(3.2.11)
or
A + ( √33 )BT = 1
σ T (3.2.12)
Next consider a uniaxial compressive stress state characterized by the following
stress tensor.
52
σ ij=[0 0 00 −σ C 00 0 0 ]
(3.2.13)
then
Sij = ¿[ 13
0 0 ¿][ 0 - 23
0 ¿]¿¿
¿
(3.2.14)
and
( I1 , √J 2 , 3√J 3) = (−σC , √33
σC ,3√23
σC )(3.2.15)
θ = 600
(3.2.16)
Substitution of these the values for the invariants into the Willam-Warnke (1975) failure
function yields
A(−σ C ) + (√33
BC)(σ C ) − 1 = 0(3.2.17)
−A + (√33 )BC = 1
σC (3.2.18)
53
1x
2x
3x
BC
BC
BCBC
At this point we have two equations (3.2.12 and 3.2.18) and three unknowns (A,
BT, and BC). In order to obtain a third equation consider an equal biaxial compressive
stress state characterized as
σ ij=[0 0 00 −σ BC 00 0 −σBC
](3.2.19)
Now the deviatoric stress tensor becomes
Sij = ¿[ 23
0 0 ¿] [ 0 - 13
0 ¿]¿¿
¿
(3.2.20)
and
( I1 , √J 2 , 3√J 3) = (−2 σBC , √33
σBC , −3√23
σBC)(3.2.21)
θ = 0o
(3.2.22)
Substitution of these invariants into failure function defined by equation (3.2.1) yields
(2 A ) (−σ BC) + (√33
BT ) (σ BC) − 1 = 0(3.2.23)
54
or
−2 A + (√33 )BT = 1
σBC (3.2.24)
We now have three equations, i.e., (3.2.12), (3.2.18) and (3.2.24), in three
unknowns A, Bt and Bc. Solution of this system of equations leads to the following three
expressions for the unknowns model parameters
A = ( 13 )( 1
σ T−
1σ C )
(3.2.25)
and
BT = (√33 )( 2
σT+ 1
σ BC )(3.2.26)
and
BC = (√3 ) [ 1σC
+ ( 13 )( 1
σT− 1
σ BC )](3.2.27)
In order to characterize to characterize the Willam and Warnke (1975) model in a
straight forward manner one would need failure data from a uniaxial load path, a uniaxial
compressive load path, and an equal biaxial compression load path. Unfortunately,
Burchell (2007) did not conduct biaxial compression stress tests. It must be pointed out
that these tests are extremely difficult to perform. Here we arbitrarily assume the
55
magnitude of the biaxial compression stress at failure is 1.16 times the uniaxial
compression stress at failure. Thus the three sets of strength parameters obtained from
the Burchell (2007) data are
T = 15.93 MPa (3.2.28)
for tension,
C = 52.93 MPa (3.2.29)
for compression and
BC = 61.40 MPa (3.2.30)
for the biaxial compression material strength. The important thing is that with the three
parameter Willam-Warnke (1975) criterion the biaxial compression strength is a direct
model input. Biaxial compression strength could be controlled indirectly in the Drucker-
Prager model (1952). The additional strength parameter in the Willam-Warnke (1975)
model brings additional flexibility and the criterion represents an increased flexibility in
modeling material behavior relative to the Drucker-Prager (1952) criterion in a manner
similar to a comparison of the Drucker-Prager (1952) model to the von Mises model
(1913). However, the additional flexibility is not enough to capture the anisotropic
behavior exhibited by Burchell’s (2007) graphite data.
This is evident in Figure 3.2.2 where the Willam-Warnke (1975) criterion and all
of Burchell’s (2007) test data are projected onto the principal stress plane defined by the
1 - 2 coordinate axes. The criterion seems to capture the biaxial failure data along load
56
path #B-8. However, there is an increasing loss of fidelity with load paths #B-7 and #B-
6. Load path #B-5 represents anisotropic strength behavior and the Willam and Warnke
(1975) model was constructed based on the assumption of an isotropic material. The
same behavior can be seen in biaxial load paths #B-4, #B-3 and #B-2. As we move away
from the load paths used to characterize the model parameters we encounter the loss in
fidelity and here we attribute the loss to material anisotropy. This anisotropic phenomena
will drive the research proposed for this effort.
IDENTIFY THE LOAD PATHS IN FIGURE 3.2.2 JUST LIKE FIGURE 2.4.1
57
MPa93.15,0
, 2211
MPaMPa 40.61,40.61
, 2211
MPa93.52,0
, 2211
1
1
)(11 MPa
)(22 MPa
Figure 3.2.2 The Willam-Warnke(1975) criterion projected onto the 1 -2 principle stress plane (T = 15.93 MPa, C = 52.93 MPa, BC = 61.40 MPa)
Varying the biaxial compressive strength of the material does not help in
matching the criterion with the data from biaxial load paths #B-2 through #B-4 and load
paths #B-6 as well as #B-7. This is evident in Figure 3.2.3 where the biaxial strength is
varied from 0.96 of the uniaxial compression strength to 1.16 times the uniaxial
58
)(11 MPa
)(22 MPa
compressive strength. The various projections based on differing values of BC did not
improve the criterion’s ability to match the data from the load paths just mentioned.
Figure 3.2.3 The Willam-Warnke(1975) criterion projected onto the 1 -2 principle stress plane for multiple T = 15.93 MPa, C = 52.93 MPa, and BC = 56.11MPa, 61.40
MPa, 66.69Mpa
In Figures 3.2.4 and 3.2.5 the Willam-Warnke (1975) model is projected onto
deviatoric planes. In Figure 3.2.4
ξ = 9 . 20 MPa (3.2.31)
and in Figure 3.2.5
59
ξ = −30 .2 MPa(3.2.32)
In these figures the Willam-Warnke (1975) criterion presents as slices through a right
triangular pyramid.
Figure 3.2.4 The Willam-Warnke criterion projected onto a deviatoric plane
(ξ=6 . 05 Mpa) parallel to the -plane with T = 15.93 MPa, C = 52.93 MPa,
BC = 61.40 MPa
60
)(MPa
)(MPar
o30
o60
o0
)0,13.50,9.70(),,( oBC MPaMPar
)60,33.43,56.30(),,( oC MPaMPar
)0,01.13,02.9(),,( oT MPaMPar
Figure 3.2.5 The Willam-Warnke criterion projected onto a deviatoric plane
(ξ1=−30 . 2 MPa) parallel to the -plane with T = 15.93 MPa, C = 52.93 MPa,
BC =61.40 MPa
Figure 3.2.6 The Willam-Warnke criterion projected onto the meridian plane for a material strength parameter of T = 15.93 MPa, C = 52.93 MPa,
BC = 61.40 MPa
The meridian lines associated with the Willam-Warnke (1975) failure criterion for
Lode angle values of θ=0oandθ=60o
are depicted on Figure 3.2.6. The meridians for
each Lode angle are distinct from one another since one is a tensile meridian and passes
through the tensile strength parameter along a principle stress axis. The other is a
61
compressive meridian and intercepts a principle stress axis at the value of the
compressive strength parameter. The θ=0omeridian line goes through point (r = 9.02
MPa, = 13.01 MPa) and the θ=60omeridian line goes through point (r = 30.56 MPa,
= 43.22 MPa) as they should since these data values were used to characterize the
model. The point (r = 6.05 MPa, = 8.56 MPa) represents the average strength for load
path #B-5, a uniaxial load path that was not used to characterize the model. The data
from load path #B-5 represents anisotropic strength behavior and one should not expect
this data to match well with the isotropic Willam-Warnke (1975) model.
Up to this point it has been noted several times that Burchell’s data (2007)
exhibits anisotropic behavior. As part of this effort many attempts were made to extend
the Willam-Warnke (1975) model in order to capture anisotropic behavior through the
use of tensor based stress invariants. The primary difficulty with extending the Willam-
Warnke (1975) failure criterion to anisotropy is the fact that the function is linear in stress
and based on a trial and error approach, the belief here is that at least a quadratic
dependence is needed in order to capture anisotropic behavior through stress invariants.
The next section presents a failure criterion analogous to the Willam-Warnke (1975)
failure model that is quadratic in stress.
62
CHAPTER IV
ISOTROPIC GREEN-MKRTICHIAN FAILURE CRITERION
The next phenomenological failure criterion considered in this effort was
proposed by Green and Mkrtichian (1977) and has the basic form
g = g (σ ij , a i )(4.1)
Green and Mkrtichian (1977) track the principle stress direction vectors, identified here
as ai. Utilizing the eigenvectors of the principal stresses enables the identification of
tensile and compressive principle stress directions. The authors of this model consider
different behavior in tension and compression as a type of material anisotropy. Utilizing
first order tensors (the eigenvectors) as directional tensors is an accepted approach in
modeling anisotropy through the use of invariants. Spencer (1984) pointed out the
mathematics that underlies the concept. This model produces results very similar to the
Willam-Warnke (1975) failure criterion. The utility of the isotropic form of the Green-
Mkrtichian (1977) model is that this failure criterion is quadratic in stress whereas the
63
Willam-Warnke (1975) failure criterion was linear in stress. Being quadratic in stress
makes the Willam-Warnke (1975) model more amenable to including anisotropic
behavior, which is discussed in the next chapter. This chapter outlines fundamental
aspects of the Willam-Warnke (1975) failure criterion in preparation for the extension to
anisotropy.
4.1 Integrity Basis and Functional Dependence
The integrity basis for the a function with a dependence specified in equation
(4.1) is
iiI 1(4.1.1)
I 2 = σ ij σ ji (4.1.2)
I 3 = σ ij σ jk σki (4.1.3)
I 4 = ai a j σ ij (4.1.4)
and
I 5 = ai a j σ jk σki (4.1.5)
These invariants (with the exception of I3, which can be derived from I1 and I2) constitute
an integrity basis and span the space of possible stress invariants that can be utilized to
64
compose scalar valued functions that are dependent on stress. Thus the dependence of
the Green and Mkrtichian (1977) failure criterion can be characterized in general as
g (σ ij ) = g ( I 1 , I 2 , I 4 , I 5)(4.1.6)
One possible polynomial formulation for g in terms of the integrity basis is
g (σ ij ) = A ( I1 )2 + B I 2 + C I 1 I 4 + D I 5 − 1(4.1.7)
This functional form is quadratic in stress which, as is seen in the next section, is
convenient when extending this formulation to include anisotropy. The invariants I4 and
I5 are associated with the directional tensor ai, and we note that Green and Mkrtichian
(1977) utilized these invariants in the functional dependence very judiciously. They
partitioned the Haigh-Westergaard stress space and offered four forms for the failure
functions.
4.2 Functional Forms and Associated Gradients by Stress Region
By definition the principal stresses are identified such that
σ 1 ≥ σ 2 ≥ σ3(4.2.1)
The four functions proposed by Green and Mkrtichian (1977) span the stress space which
is partitioned as follows:
Region #1: σ 1 ≥ σ 2 ≥ σ3 ≥ 0 – all principal stresses are tensile
65
Region #2: σ 1 ≥ σ 2 ≥ 0 ≥ σ3 – one principal stress is compressive, the
others are tensile
Region #3: σ 1 ≥ 0 ≥σ2 ≥ σ3 – one principal stress is tensile, the others are
compressive
Region #4: 0 ≥ σ1 ≥ σ2 ≥ σ3 – all principal stresses are compressive.
Thus the Green and Mkrtichian (1977) criterion has a specific formulation for the case of
all tensile principle stresses, and a different formulation for all compressive principle
stresses (see derivation below). For these two formulations there is no need to track
principle stress orientations and thus for Regions #1 and #4 the Green and Mkrtichian
(1977) failure criterion did not include the terms associated with I4 and I5, both of which
contain information regarding the directional tensor. A third and fourth formulation
exists for Regions #3 and #4 where two principle stresses are tensile and when two
principle stresses are compressive, respectively and the failure behavior depends on the
direction of the principal tensile and compressive stresses. For these regions of the stress
space for the Green and Mkrtichian (1977) failure criterion includes the invariants I4 and
I5.
The functional values of the four formulations g1, g2, g3 and g4 must match along
their common boundaries. In addition, the tangents associated with the failure surfaces
66
along the common boundaries must be single valued. This will provide a smooth
transition from one region to the next. To insure this, the gradients to the failure surfaces
along each boundary are equated. The specifics of equating the formulations and
equating the gradients at common boundaries are presented below. Relationships are
developed for the constants associated with each term of the failure function for the four
different regions.
Region #1: (σ1 ≥ σ2 ≥ σ3 ≥0 ) Green and Mkrtichian (1977) assumed
the failure function for this region of the stress space is
g1 = 1 − [( 12 ) A1 I
12 + B 1 I 2 ](4.2.2)
From equation (4.2.10) it is evident that there will be a group of constants for each region
of the stress space. Hence the subscripts for the constants associated with each invariant
as well as the failure function will run from one to four. Also note the absence of
invariants I4 and I5. The corresponding normal to the failure surface is
∂ g1
∂σ ij=
∂ g1
∂ I 1
∂ I 1
∂ σ ij+
∂ g1
∂ I 2
∂ I 2
∂ σ ij (4.2.3)
where
∂ g1
∂ I1= −A1 I 1
(4.2.4)
67
∂ g1
∂ I2= −B1
(4.2.5)
∂ I 1
∂σ ij= δij
(4.2.6)
and
∂ I 2
∂σ ij= 2 σ ij
(4.2.7)
Here ij is the Kronecker delta tensor. Substitution of equations (4.2.4) through (4.2.7)
into (4.2.3) leads to the following tensor expression
∂ g1
∂σ ij= −( A1 I 1 δij + 2 B1 σ ij )
(4.2.8)
or in a matrix format
∂ g1
∂ σ ij= [σ1+σ2+σ 3 0 0
0 σ 1+σ 2+σ 3 00 0 σ1+σ2+σ3
] (−A1)
+ [2 σ1 0 00 2σ2 00 0 2σ3
] (−B1 )
(4.2.9)
The matrix formulation allows easy identification of relationships between the various
constants.
Region #2: (σ1 ≥ σ2 ≥ 0 ≥ σ 3) The failure function for region #2 is
68
g2 = 1 − [( 12 ) A2 I
12 + B 2 I2 + C2 I 1 I 4 + D2 I 5 ] (4.2.10)
Note the subscripts on the constants and the failure function. The normal to the surface is
∂ g2
∂σ ij=
∂ g2
∂ I 1
∂ I 1
∂ σ ij+
∂ g2
∂ I2
∂ I 2
∂σ ij+
∂ g2
∂ I 4
∂ I 4
∂ σ ij+
∂ g2
∂ I 5
∂ I5
∂ σ ij (4.2.11)
Here
∂ g2
∂ I1= −( A2 I 1 + C2 I 4 )
(4.2.12)
∂ g2
∂ I 2= −B2
(4.2.13)
∂ g2
∂ I 4= −C2 I 1
(4.2.14)
∂ g2
∂ I5= −D2
(4.2.15)
∂ I 4
∂σ ij= ai a j
(4.2.16)
and
∂ I 5
∂σ ij= ak a i σ jk + a j ak σ ki
(4.2.17)
69
The principle stress direction of interest in this region of the stress space is the one
associated with the third principal stress. Assuming the Cartesian coordinate system is
aligned with the principal stress directions then the eigenvector associated with the third
principal stress is
a i = (0 , 0 , 1 )(4.2.18)
Thus for equation (4.2.16) and (4.2.17)
ak a i = a j ak = ai a j = [001 ] [0 0 1 ]
¿ [0 0 00 0 00 0 1 ]
(4.2.19)
Given the principle stress direction of interest the fourth and fifth invariants are
I 4 = σ 3(4.2.20)
and
I 5 = σ32
(4.2.21)
for this region of the stress space. Substitution of equations (4.2.12) through (4.2.21) into
(4.2.11) yields the following tensor expression for the normal to the failure surface
∂ g2
∂ σ ij= −[ A2 I1 δij + 2B2 σ ij + C2( σ3 δ ij + I1 ai ai )
+ D2 (ak ai σ jk + a j ak σ ki) ] (4.2.22)
70
The matrix form of equation (4.2.22) is
∂ g2
∂ σ ij= [σ1+σ2+σ 3 0 0
0 σ 1+σ 2+σ 3 00 0 σ1+σ2+σ3
] (−A2) + [2 σ1 0 00 2σ 2 00 0 2σ3
] (−B2)
+ [σ 3 0 00 σ 3 00 0 σ 1+σ 2+2 σ3
] (−C2 ) + [0 0 00 0 00 0 2 σ3
] (−D2)
(4.2.23)
Region #3:(σ1 ≥ 0 ≥σ2 ≥ σ 3) The failure function for this region of the
stress space is
g3 = 1 − [(12 ) A3 I
12 + B 3 I 2 + C3 I1 I 4 + D3 I 5] (4.2.24)
The normal to the surface is
∂ g3
∂σ ij=
∂ g3
∂ I 1
∂ I 1
∂σ ij+
∂ g3
∂ I 2
∂ I 2
∂ σ ij+
∂ g3
∂ I 4
∂ I 4
∂ σ ij+
∂ g3
∂ I5
∂ I 5
∂σ ij (4.2.25)
Here
∂ g3
∂ I 1= −( A3 I1 + C3 I 4 )
(4.2.26)
∂ g3
∂ I 2= −B3
(4.2.27)
∂ g3
∂ I 4= −C3 I1
(4.2.28)
71
and
∂ g3
∂ I5= −D3
(4.2.29)
The principle stress direction of interest for this stress state is the one associated with the
first principal stress, i.e.,
a i = (1 , 0 , 0 )(4.2.30)
Now
ak a i = a j ak = ai a j = [100 ] [1 0 0 ]
¿ [1 0 00 0 00 0 0 ]
(4.2.31)
The fourth and fifth invariants for this region of the stress space are
I 4 = σ1(4.2.32)
and
I 5 = σ12
(4.2.33)
Substitution of the quantities specified above into (4.2.25) yields the following tensor
expression
72
∂ g3
∂ σ ij= −[ A3 I1 δij + 2 B3 σ ij + C3 (σ1 δ ij + I 1 ai a j )
+ D3( ak a i σ jk + a j ak σki )] (4.2.34)
The matrix form of this equation is as follows
∂ g3
∂ σ ij= [σ1+σ2+σ 3 0 0
0 σ 1+σ 2+σ 3 00 0 σ1+σ2+σ3
] (−A3 ) + [2 σ1 0 00 2σ 2 00 0 2σ3
] (−B3)
+ [2 σ1+σ2+σ3 0 00 σ1 00 0 σ1
] (−C3) + [2 σ1 0 00 0 00 0 0 ](−D3)
(4.2.35)
Region #4: 0 ≥ σ1 ≥ σ2 ≥ σ3 The failure function for this region of the
stress space is
g4 = 1 − [( 12 ) A4 I
12 + B 4 I2 ](4.2.36)
The corresponding normal to the failure surface is
∂ g4
∂σ ij=
∂ g4
∂ I 1
∂ I1
∂ σ ij+
∂ g4
∂ I2
∂ I 2
∂ σ ij (4.2.37)
Here
∂ g4
∂ I 1= A4 I 1
(4.2.38)
and
73
∂ g4
∂ I 2= B4
(4.2.39)
Substitution of the equations above into (4.2.37) yields the following tensor expression
∂ g4
∂σ ij= 1 − ( A4 I 1 δij + 2 B4 σ ij )
(4.2.40)
The matrix format of this expression is
∂ g4
∂ σ ij= [σ1+σ2+σ3 0 0
0 σ1+σ2+σ 3 00 0 σ 1+σ 2+σ 3
](− A4)
+ [2 σ1 0 00 2 σ2 00 0 2σ 3
](−B4 )
(4.2.41)
4.3 Relationships Between Functional Constants
With the failure functions and the normals to those functions defined for each
region, attention is now turned to defining the constants. Consider the region of the
Haigh-Westergaard stress space where with andThe stress
state in a matrix format is
σ ij = ¿ [σ1 0 0 ¿ ] [0 σ2 0 ¿ ]¿¿
¿(4.3.1)
74
and this stress state lies along the boundary shared by region #1 and region #2. At this
boundary we impose
g1 = g2 (4.3.2)
and
∂ g1
∂σ ij=
∂ g2
∂ σ ij (4.3.3)
For this stress state the invariants I1 and I2 are
I 1 = σ1 + σ 2(4.3.4)
and
I 2 = σ12 + σ
22
(4.3.5)
for both g1 and g2. The invariants I4 and I5 for g2 are
I 4 = σ 3 = 0(4.3.6)
and
I 5 = (σ3 )2 = 0(4.3.7)
Substitution of equations (4.3.4) through (4.3.7) into equation (4.3.1) yields the following
matrix expression
75
[σ1+σ2+σ3 0 00 σ1+σ2+σ3 00 0 σ1+σ 2+σ 3
]A1 + ¿ [2 σ1 0 0 ¿ ] [0 2 σ2 0 ¿ ] ¿¿
¿
¿
¿
(4.3.8)
with
σ 3 = 0(4.3.9)
then
[σ1+σ2 0 00 σ1+σ2 00 0 σ1+σ 2
] A1 + ¿ [2σ1 0 0¿ ] [ 0 2σ2 0¿ ]¿¿
¿
¿
¿
(4.3.10)
The following three expressions can be extracted from equation (4.3.10)
(σ1+σ2 ) A1 + (2 σ 1)B1 = (σ1+σ2 ) A2 + (2 σ1 )B2(4.3.11)
(σ1+σ2 ) A1 + (2 σ 2) B1 = (σ1+σ2 ) A2 + (2 σ 2)B2(4.3.12)
76
1x
2x
3x
2
23
3
(σ1+σ2 ) A1 = (σ1+σ2 ) A2 + (σ1+σ2 )C2(4.3.13)
The constant D2 does not appear due to its multiplication with the null matrix. However,
C2 does appear in the third expression but in the first two immediately above. Focusing
on equation (4.3.11) and equation (4.3.12) which represents two equations in two
unknowns then
B1 = B2(4.3.14)
and
A1 = A2(4.3.15)
Substitution of equation (4.3.15) into equation (4.3.13) yields
C2 = 0(4.3.16)
and at this point D2 is indeterminate. Substitution of equations (4.3.4) through (4.3.7)
into equation (4.3.2) yields the following expression
Now consider the region of the Haigh-Westergaard stress space where
and The stress state in a matrix format is
σ ij = ¿ [σ1 0 0 ¿ ] [ 0 0 0 ¿ ]¿¿
¿(4.3.17)
77
and this stress state lies at the boundary shared by region #2 and region #4. At this
boundary we impose
g2 = g3(4.3.18)
and
∂ g2
∂σ ij=
∂ g3
∂ σ ij (4.3.19)
Under these conditions the invariants I1 and I2 are
I 1 = σ1 + σ 3(4.3.20)
and
I 2 = σ12 + σ
32(4.3.21)
Substitution of equation (4.3.20) and (4.3.21) into equation (4.3.19) yields
¿ [ σ1+σ 2+σ 3 0 0¿ ] [ 0 σ1+σ2+σ3 0¿ ]¿¿
¿
¿¿
(4.3.22)
78
with
σ 2 = 0(4.3.23)
then
¿ [ σ1+σ 3 0 0 ¿ ] [ 0 σ1+σ3 0¿ ] ¿¿
¿
¿¿
(4.3.24)
The following three expressions can be extracted from equation (4.3.24)
(σ1+σ3 ) A2 + (2 σ1) B2 + ( σ3 )C2
¿ (σ1+σ3 ) A3 + (2 σ1) B3 + ( 2σ1+σ 3)C3 + (2 σ1 ) D3 (4.3.25)
(σ1+σ3 ) A2 + ( σ3 )C2 = ( σ1+σ 3 ) A3 + (σ 1) C3(4.3.26)
and
(σ1+σ3 ) A2 + (2σ3) B2 + (σ1+2σ 3) C2 + (2σ3 ) D2
¿ (σ1+σ3) A3 + (2 σ3) B3 + (σ1 ) C3 (4.3.27)
Earlier it was determined that C2 = 0, so from equation (4.3.26) we obtain
79
A2 = A3 +σ3
(σ1+σ3 )C3
(4.3.28)
From equation (4.3.74) and equation (4.3.75) we obtain
B2 = B3 +(σ1+σ3)
2 σ1C3 + D3
(4.3.29)
In addition, from equation (4.3.27) and equation (4.3.26) we obtain
D2 = −(σ1+σ3 )
2 σ1C3 − D3
(4.3.30)
Consider the Region of the Haigh-Westergaard stress space where
and The stress state in a matrix format is
σ ij = ¿ [ 0 0 0 ¿ ] [0 σ 2 0¿ ]¿¿
¿(4.3.31)
and this stress state lies along the boundary shared by region #3 and region #4. At this
boundary we impose
g3 = g4(4.3.32)
and
∂ g3
∂σ ij=
∂ g4
∂ σ ij (4.3.33)
Under these conditions the invariants I1 and I2 are
80
I 1 = σ2 + σ 3(4.3.34)
and
I 2 = σ22 + σ
32(4.3.35)
Substitution of equations (4.3.34) and (4.3.35) into equation (4.3.32) yields
[σ1+σ2+σ3 0 00 σ1+σ 2+σ 3 00 0 σ 1+σ 2+σ3
]A3 + [2 σ 1 0 00 2σ2 00 0 2 σ 3
]B3
+ [2 σ1+σ2+σ3 0 00 0 00 0 0 ]C3 + [2 σ1 0 0
0 0 00 0 0 ] D3
¿ [σ1+σ2+σ3 0 00 σ1+σ2+σ3 00 0 σ1+σ2+σ 3
] A4
+ [2σ 1 0 00 2σ2 00 0 2σ 3
]B4
(4.3.36)
with
σ 1 = 0(4.3.37)
then
81
¿ [ σ2+σ 3 0 0 ¿ ] [ 0 σ2+σ3 0 ¿ ] ¿¿
¿
¿¿
(4.3.38)
The following three expressions can be extracted from equation (4.3.38)
(σ 2+σ3 ) A3 + (σ2+σ3 ) C3 = (σ2+σ3 ) A4(4.3.39)
(σ 2+σ3 ) A3 + (2 σ2) B3 = ( σ2+σ3 ) A4 + (2 σ2) B4(4.3.40)
and
(σ 2+σ3 ) A3 + (2 σ3 )B3 = (σ2+σ3 ) A4 + (2 σ3 )B4(4.3.41)
From equation (4.3.39) we discern that
A3 + C3 = A4(4.3.42)
From equation (4.3.40) and equation (4.3.41) we obtain
(2 σ 2−2 σ 3) B3 = (2 σ2−2 σ3 )B4(4.3.43)
or that
82
B3 = B4(4.3.44)
Substitution of equations (4.2.34) and (4.2.35) into equation (4.3.41) leads to
A3 = A4(4.3.45)
and
C3 = 0(4.3.46)
Substitution of equation (4.3.46) into equations (4.3.28), (4.3.29) and (4.3.30) yields
A2 = A3(4.3.47)
B2 = B3 + D3(4.3.48)
and
D2 = − D3(4.3.49)
So the relationships between the functional constants is as follows
A1 = A2 = A3 = A4(4.3.50)
B1 = B2= B3 − D2= B4 − D2(4.3.51)
C2 = C3 = 0(4.3.52)
and
D2 + D3 = 0(4.3.53)
83
1x
2x
3x
T
T
These relationships insure that the four functional forms for the failure function are
smooth and continuous along the boundaries of the four regions.
4.4 Functional Constants in Terms of Strength Parameters
Next we utilize specific load paths in order to define the constants defined above
in terms of stress values obtained at failure. Consider the following stress state at failure
under a uniaxial tensile load
σ ij = [0 0 00 σT 00 0 0 ]
(4.4.1)
This stress state lies on the boundary of region #1. The invariants for this stress state are
I 1 = σT(4.4.2)
and
I 2 = σT2
(4.4.3)
The failure function takes the form
g1 = 1 − [12 A1 (σT )2 + B1 (σT )2 ]¿ 0
(4.4.4)
84
1x
2x
3x
C
C
from which the following relationship is obtained
12
A1 + B1 = 1σ
T2(4.4.5)
Next a uniaxial compressive stress state is considered where
σ ij = [0 0 00 0 00 0 σ C
](4.4.6)
The principle stress direction for this stress state is
a i = (0 , 0 , 1 )(4.4.7)
thus
a i a j = [0 0 00 0 00 0 1 ]
(4.4.8)
The invariants are as follows
I 1 = −σ C(4.4.9)
I 2 = (−σC )2(4.4.10)
85
1x
2x
3x
BC
BC
BCBC
I 4 = σC(4.4.11)
and
I 5 = (−σC )2(4.4.12)
Thus
g2 = 12
A2 ( σC )2 + B2 (σC )2 + D2 (σC )2 −1
¿ 0(4.4.13)
which leads to
12
A2 + B2 + D2 = 1σ
C2(4.4.14)
Next consider an equal biaxial compressive stress state where
σ ij = [0 0 00 −σ BC 00 0 −σ BC
](4.4.15)
86
This stress state lies within region #4 and the invariants are as follows
I 1 = −2 σ BC(4.4.16)
and
I 2 = σBC 2
(4.4.17)
The failure function for this particular stress state is
f 4 = 12
A4 (2σ BC )2 + B4 (2 σBC2) − 1
¿ 0(4.4.18)
which leads to
A1 + B1 + D2 = 12 (σ BC )2
(4.4.19)
Solving equations (4.4.5), (4.4.4) and (4.4.19) using equations (4.3.50) through (4.3.53)
leads to
A1 = A2 = A3 = A4 = 1σ
BC2− 2
σC2
(4.4.20)
B1 = B2 = 1σ
T2− 1
2 σBC2
+ 1σ
C2(4.4.21)
B3 = B4 = 2σ
C2− 1
2 σBC 2
(4.4.22)
and
87
D2 = −D3 = 1σ
C2− 1
σT2
(4.4.23)
In order to visualize the isotropic version of the Green and Mkrtichian (1977)
failure criterion relative to Burchell’s (2007) failure data, values were computed for the
strength constants immediately above, i.e., T = 15.93 MPa for tension, C = 52.93 MPa
for compression and BC = 61.40 MPa for the biaxial compression. The values for T and
C and were obtained directly from Burchell’s (2007) data. The value for BC was
determined by a best fit approximation of the failure curve to the data in Figure 4.4.1.
Various projections of the isotropic Green and Mkrtichian (1977) failure criterion are
presented in the next several figures along with the Burchell (2007) data. The first is a
projection onto the 11 – 22 stress space which is depicted in Figure 4.4.1. As can be
seen in this figure the isotropic Green and Mkrtichian (1977) model captures the different
behavior in tension and compression exhibited by the Burchell (2007) data along the 22
axis. However, the Green and Mkrtichian (1977) failure criterion does not capture
material anisotropy which is clearly exhibited by the Burchell (2007) failure data along
the tensile segments of the 11 axis relative to the 22 axis.
88
MPa93.15,0
, 2211
MPaMPa 40.61,40.61
, 2211
MPa93.52,0
, 2211
1
1
)(11 MPa
)(22 MPa
Figure 4.4.1 The Green-Mkrtichian criterion projected onto the 1 -2 principle stress plane (T= 15.93 MPa, C = 52.93 MPa, BC = 61.40 MPa)
The isotropic Green and Mkrtichian (1977) failure criterion is projected onto the
deviatoric planes in Figures 4.4.2 and 4.4.3. Note that a cross section through the failure
function perpendicular to the hydrostatic axis transitions from a pyramidal shape (Figure
4.4.3) to a circular shape (Figure 4.4.2) with an increasing value of the stress invariant I1.
This suggests that the apex of the failure function presented in a full Haigh-Westergaard
stress space is blunt, i.e., quite rounded for the this particular criterion.
89
Figure 4.4.2 The Green-Mkrtichian criterion projected onto a deviatoric plane
(ξ=9. 20 Mpa) parallel to the -plane with T = 15.93 MPa, C = 52.93 MPa,
BC = 61.40 MPa
90
)(MPa
)(MParo30
o60
o0)0,13.50,9.70(),,( o
BC MPaMPar
)60,33.43,56.30(),,( oC MPaMPar
)0,01.13,02.9(),,( oT MPaMPar
Figure 4.4.3 The Green-Mkrtichian criterion projected onto a deviatoric plane
(ξ=−30 . 2MPa ) parallel to the -plane with T = 15.93 MPa, C = 52.93MPa,
BC = 61.40 MPa
The meridian lines of the isotropic Green-Mkrtichian (1977) failure surface
corresponding to θ=0oandθ=60o
are depicted on Figure 4.4.4. Obviously the
meridian lines are not linear. The θ=0omeridian line goes through point defined by
9.02 MPa and r = 13.01 MPa. The θ=60omeridian line goes through the point
defined by = 30.56 MPa and r = 43.22 MPa.
91
Figure 4.4.4 The Green-Mkrtichian criterion projected onto the meridian plane for a material strength parameter of T = 15.93 MPa, C = 52.93 MPa, BC = 61.40 MPa
As the value of the I1 stress invariant associated with the hydrostatic stress
increases in the negative direction, failure surfaces perpendicular to the hydrostatic stress
line become circular again. The model suggests that as hydrostatic compression stress
increases the difference between tensile strength and compressive strength diminishes
and approach each other asymptotically. This is a material behavior that should be
verified experimentally in a manner similar to Bridgman’s (1953) bend bar experiments
conducted in hyperbaric chambers on cast metal alloys. Balzer (1998) provides an
excellent overview of Bridgman’s experimental efforts, as well as others and their
accomplishments in the field of high pressure testing.
However, as indicated in Figure 4.4.1, the isotropic formulation of the Green-
Mkrtichian (1977) failure criterion does not capture the anisotropic behavior of
Burchell’s (2007) data. The isotropic formulation is extended to transverse isotropy in
the next chapter. Orthotropic behavior and other types of anisotropic behavior can be
captured through similar use of tensorial invariants.
92
CHAPTER V
ANISOTROPIC GREEN-MKRTICHIAN MODEL
As discussed in earlier sections the Burchell (2007) multiaxial failure data
strongly suggests that the graphite tested was anisotropic. Thus there is a need to extend
the isotropic failure model discussed in the previous section so that anisotropic failure
behavior is captured. This can be done again by utilizing stress based invariants where
the material anisotropy is captured through the use of a direction vector associated with
primary material directions. The concept is identical to the extension of the isotropic
inelastic constitutive model . The extension of a phenomenological failure criterion will
be made for a transversely isotropic material. Other material symmetries, e.g., an
orthotropic material symmetry, can be included as well. Duffy and Manderscheid
(1990b) as well as others have suggested an appropriate integrity basis for the orthotropic
material symmetry. Transversely isotropic materials have the same properties in one
plane and different properties in a direction normal to this plane. Orthotropic materials
have different properties in three mutually perpendicular directions.
93
5.1 Integrity Base for Anisotropy
The preferred material direction is designated through a second direction vector,
di. The dependence of the failure function is extended such that
g (σ ij , d i d j , a i a j ) = 0 (5.1.1)
The definition of the unit vector ai is the same as in earlier sections. Rivlin and
Smith (1969) as well as Spencer (1971) show that for a scalar valued function with
dependence stipulated by equation (5.1.1) the integrity basis is
I 1 = σkk (5.1.2)
I 2 = σ ij σ ji (5.1.3)
I 3 = σ ij σ jk σ ki (5.1.4)
I 4 = ai a j σ ij (5.1.5)
I 5 = ai a j σ jk σki (5.1.6)
I 6 = d i d j σ ji (5.1.7)
I 7 = d i d j σ jk σki (5.1.8)
I 8 = ai a j d j dk σ kj (5.1.9)
and
94
I 9 = ai a j d j dk σ km σ mi (5.1.10)
The invariant I3 is omitted again since this invariant is cubic in stress. As before those
invariants linear in stress enter the functional dependence as squared terms or as products
with another invariant linear in stress. Therefore the anisotropic failure function has the
following dependence
g (σ ij , d i d j , a i a j ) = g ( I 1 , I 2 , I 4 , I 5 , I6 , I7 , I 8 , I 9 )¿ 1 − [ (1/2 ) AI
12 + BI 2 + CI 1 I 4 + DI 5
+ EI 1 I 6 + FI7 + GI 1 I 8 + HI9 ] (5.1.11)
The form of the failure function was constructed as a polynomial in the invariants listed
above. The constants in this formulation (A, B, C, D, E, F, G, and H) are characterized
by adopting simple strength tests. The proposed failure function was incorporated into a
reliability model through the use of Monte Carlo simulation and importance sampling
techniques. This feature is discussed in a subsequent section.
5.2 Functional Forms and Associated Gradients by Stress Region
Similar to the approach adopted for anisotropic constitutive models, the
underlying concept is that the response of the material depends on the stress state, a
preferred material direction and whether the principal stresses are tensile or compressive.
The principle stress space is divided again into four regions. The regions and associated
95
failure functions are listed below. In the first region all of the principle stresses are
tensile, i.e.,
Region #1: (σ1 ≥ σ2 ≥ σ3 ≥0 )
g1 = 1 − [( 12 ) A1 I
12 + B 1 I 2 + E 1 I 1 I 6 + F 1 I 7] (5.2.1)
In Region #1 a direction vector associated with the principle stresses is unnecessary since
all principle stresses are tensile. The corresponding normal to the failure surface is
∂ g1
∂σ ij=
∂ g1
∂ I 1
∂ I 1
∂ σ ij+
∂ g1
∂ I 2
∂ I 2
∂ σ ij+
∂ g1
∂ I6
∂ I 6
∂σ ij+
∂ g1
∂ I7
∂ I 7
∂σ ij (5.2.2)
where
∂ g1
∂ I1= −A1 I 1 − E1 I 6
(5.2.3)
∂ g1
∂ I2= −B1
(5.2.4)
∂ g1
∂ I6= −E1 I 1
(5.2.5)
∂ g1
∂ I7= −F1
(5.2.6)
∂ I 1
∂σ ij= δij
(5.2.7)
96
∂ I 2
∂σ ij= 2 σ ij
(5.2.8)
∂ I 6
∂σ ij= di d j
(5.2.9)
and
∂ I 7
∂σ ij= dk d j σki + d i dk σ jk
(5.2.10)
Substitution of equations (5.2.3) through (5.2.10) into (5.2.2) leads to the following
tensor expression
∂ g1
∂ σ ij= −[ ( A1 I1 + E1 I6 ) δ ij + 2 B1 σ ij + E1 I 1 d i d j
+ F1(dk d i σ jk + d j d k σki ) ] (5.2.11)
or
∂ g1
∂ σ ij= −[ A1 I1 δij + 2 B1 σ ij + E1 ( I1 di d j + I 6 δ ij )
+ F1(dk d i σ jk + d j dk σki ) ] (5.2.12)
Region #2: (σ1 ≥ σ2 ≥ 0 ≥ σ 3)
g2 = 1 − [ (1/2 ) A2 I12 + B 2 I 2 + C2 I1 I4 + D2 I 5
+ E2 I 1 I 6 + F2 I7 + G2 I 1 I8 + H2 I9 ] (5.2.13)
97
In Region #2 the direction vector ai is associated with the compressive principle stress σ3.
Thus for this region
a i = (0 , 0 , 1 ) (5.2.14)
.The corresponding normal to the failure surface is
∂ g2
∂ σ ij=
∂ g2
∂ I 1
∂ I 1
∂σ ij+
∂ g2
∂ I 2
∂ I 2
∂ σ ij+
∂ g2
∂ I 4
∂ I 4
∂ σ ij+
∂ g2
∂ I 5
∂ I5
∂σ ij
+∂ g2
∂ I 6
∂ I 6
∂σ ij+
∂ g2
∂ I 7
∂ I 7
∂ σ ij+
∂ g2
∂ I 8
∂ I 8
∂ σ ij+
∂ g2
∂ I 9
∂ I 9
∂ σ ij (5.2.15)
where
∂ g2
∂ I1= −( A2 I 1 + C2 I 4 + E2 I 6 + G2 I8 )
(5.2.16)
∂ g2
∂ I 2= −B2
(5.2.17)
∂ g2
∂ I 4= −C2 I 1
(5.2.18)
∂ g2
∂ I5= −D2
(5.2.19)
∂ g2
∂ I6= −E2 I 1
(5.2.20)
∂ g2
∂ I7= −F2
(5.2.21)
98
∂ g2
∂ I8= −G2 I1
(5.2.22)
∂ g2
∂ I 9= −H2
(5.2.23)
∂ I 4
∂σ ij= ai a j
(5.2.24)
∂ I 5
∂σ ij= ak a i σ jk + a j ak σ ki
(5.2.25)
∂ I 8
∂σ ij= ai ak dk d j
(5.2.26)
and
∂ I 9
∂σ ij= ak a l d l d j σki + ai a l d l dk σ jk
(5.2.27)
Substitution of equations (5.2.7) through (5.2.10) and (5.2.16) through (5.2.27) into
(5.2.15) leads to the following tensor expression
99
∂ g2
∂ σ ij= −[ ( A2 I1 + C2 I 4 + E2 I 6 + G2 I 8) δij + 2 B2 σ ij
+ (C2 I 1) ai a j+ D2 (ak ai σ jk + a j ak σ ki )+ ( E2 I 1 ) d i d j + F2 (dk d j σki + d i dk σ jk )+ (G2 I 1 ) ai ak dk d j + H2 ( ak al d l d j σki + a i al d l dk σ jk ) ] (5.2.28)
then
∂ g2
∂ σ ij= −[ A2 I 1δ ij + 2 B2 σ ij + C2( I1 ai a j + I 4 δij )
+ D2 (ak ai σ jk + a j ak σki ) + E2 ( I 1 d i d j + I 6 δij )+ F2 (dk d j σ ki + d i dk σ jk )+ G2 ( I 1 ai ak dk d j + I 8δ ij )
+ H2 (ak a l d l d j σ ki + ai al dl dk σ jk ) ] (5.2.29)
Region #3: (σ1 ≥ 0 ≥σ2 ≥ σ 3)
The failure function for this region of the stress space is
g3 = 1 − [ (1/2 ) A3 I12 + B 3 I 2 + C3 I 1 I 4 + D3 I5 I 7
+ E3 I 1 I6 + F3 + G3 I1 I 8 + H 3 I 9 ] (5.2.30)
In Region #3 the direction vector ai is associated with the tensile principle stress direction
σ1. For this region
a i = (1 , 0 , 0 ) (5.2.31)
.The corresponding normal to the failure surface is
∂ g3
∂ σ ij=
∂ g3
∂ I 1
∂ I 1
∂ σ ij+
∂ g3
∂ I 2
∂ I 2
∂ σ ij+
∂ g3
∂ I 4
∂ I 4
∂σ ij+
∂ g3
∂ I 5
∂ I 5
∂ σ ij
+∂ g3
∂ I 6
∂ I 6
∂ σ ij+
∂ g3
∂ I 7
∂ I 7
∂ σ ij+
∂ g3
∂ I 8
∂ I 8
∂σ ij+
∂ g3
∂ I 9
∂ I 9
∂ σ ij (5.2.32)
100
where
∂ g3
∂ I 1= −( A3 I1 + C3 I 4 + E3 I 6 + G3 I 8 )
(5.2.33)
∂ g3
∂ I 2= −B3
(5.2.34)
∂ g3
∂ I 4= −C3 I1
(5.2.35)
∂ g3
∂ I5= −D3
(5.2.36)
∂ g3
∂ I6= −E3 I 1
(5.2.37)
∂ g3
∂ I7= −F3
(5.2.38)
∂ g3
∂ I 8= −G3 I 1
(5.2.39)
and
∂ g3
∂ I 9= −H3
(5.2.40)
Substitution of of equations (5.2.7) through (5.2.10), (5.2.16) through (5.2.27) and
(5.2.33) through (5.2.40) into (5.2.32) leads to the following tensor expression
101
∂ g3
∂ σ ij= −[ ( A3 I 1 + C3 I 4 + E3 I 6 + G3 I 8 ) δ ij + 2 B3 σ ij
+ (C3 I 1 )ai a j+ D3 (ak ai σ jk + a j ak σki )+ (E3 I1 ) d i d j + F3 (dk d j σ ki + di dk σ jk )+ (G3 I1 ) ai ak dk d j + H3 (ak al d l d j σ ki + ai al d l dk σ jk ) ] (5.2.41)
or
∂ g3
∂ σ ij= −[ A3 I 1 δij + 2 B3 σ ij + C3 ( I 1 ai a j + I 4 δij )
+ D3 (ak a i σ jk + a j ak σki ) + E3 ( I 1 d i d j + I6 δij )+ F3 (dk d j σ ki + d i dk σ jk )+ G3 ( I 1 ai ak dk d j + I 8 δ ij )
+ H3 ( ak a l d l d j σki + ai a l d l dk σ jk ) ] (5.2.42)
Region #4: 0 ≥ σ1 ≥ σ2 ≥ σ3
The failure function for this region of the stress space is
g4 = 1 − [ (1/2 ) A4 I12 + B4 I 2 + E4 I 1 I6 + F4 I 7]
(5.2.43)
and since all principle stresses are compressive a direction vector associated with the
principle stress direction is unnecessary. The corresponding normal to the failure surface
is
∂ g4
∂σ ij=
∂ g4
∂ I 1
∂ I1
∂ σ ij+
∂ g4
∂ I2
∂ I 2
∂ σ ij+
∂ g4
∂ I 6
∂ I 6
∂σ ij+
∂ g4
∂ I 7
∂ I 7
∂σ ij (5.2.44)
where
∂ g4
∂ I 1= − A4 I 1 + E4 I 6
(5.2.45)
102
∂ g4
∂ I 2= −B4
(5.2.46)
∂ g4
∂ I 6= −E4 I 1
(5.2.47)
∂ g4
∂ I 7= −F4
(5.2.48)
and
∂ g4
∂ I 7= −F4
(5.2.49)
Substitution of equations (5.2.7) through (5.2.10) and (5.2.45) through (5.2.49) into
(5.2.44) leads to the following tensor expression
∂ g4
∂ σ ij= −[ ( A1 I1 + E1 I6 ) δ ij + 2 B1 σ ij + E1 I 1 d i d j
+ F1(dk d i σ jk + d j d k σki ) ] (5.2.50)
or
∂ g4
∂ σ ij= −[ A1 I1 δij + 2 B1 σ ij + E1 ( I1 di d j + I 6 δ ij )
+ F1(dk d i σ jk + d j dk σki ) ] (5.2.51)
103
1x
2x
3x
YT
YT
id
5.3 Relationships the Between Functional Constants
With the failure functions and the normals to those functions defined in general
terms for each region, attention is now turned to establishing functional relationships
between the constants. Consider the following stress state at failure under a tensile load
in the preferred material direction with material direction di = (0, 1, 0), i.e.,
σ ij = [0 0 00 σYT 00 0 0 ]
(5.3.1)
The first stress subscript Y denotes a strength parameter associated with the strong
direction, and second subscript T denotes this quantity is a tensile strength
parameterThe principle stresses for this stress state are
(σ1 , σ2 , σ 3) = (σYT , 0 , 0 )(5.3.2)
and this stress state lies along the boundary shared by region #1 and region #2, as well as
the shared boundary along region #2 and region #3. At both boundaries we impose the
requirements that the gradients match, i.e.,
∂ g1
∂σ ij=
∂ g2
∂ σ ij (5.3.3)
104
∂ g2
∂σ ij=
∂ g3
∂ σ ij (5.3.4)
providing a smooth transition from one principle stress region to another. For this stress
state the first, second, sixth and seventh invariants of stress are
I 1 = σYT (5.3.5)
I 2 = σYT 2
(5.3.6)
I 6 = σYT (5.3.7)
and
I 7 = σYT 2
(5.3.8)
These stress invariants are common for stress region #1, #2 and #3. With these invariants
∂ g1
∂ σ ij= −[σYT 0 0
0 σYT 00 0 σYT
] A1 − [0 0 00 2σ YT 00 0 0 ]B1
− [σYT 0 00 2 σYT 00 0 σ YT
] E1 − [0 0 00 2 σYT 00 0 0 ]F1
(5.3.9)
For the stress state in region #2 given above the unit principle stress vector is
2 ai = (0 , 0 , 1 )(5.3.10)
The left superscript “2” denotes a vector associated the region #2. Thus
105
(2 ai ) ( 2 a j) = [0 0 00 0 00 0 1 ]
(5.3.11)
The stress invariants associated with this vector and the stress state given above are
2 I 4 = 0(5.3.12)
2 I5 = 0(5.3.13)
2 I 8 = 0(5.3.14)
and
2 I 9 = 0(5.3.15)
With these stress invariants and stress state the gradient along the boundary for stress
region #2 is
∂ g2
∂ σ ij= −[σYT 0 0
0 σ YT 00 0 σYT
] A2 − [0 0 00 2σ YT 00 0 0 ]B2 − [0 0 0
0 0 00 0 σYT
]C2
− [σYT 0 00 2σYT 00 0 σ YT
] E2 − [0 0 00 2σYT 00 0 0 ] F2
(5.3.16)
Utilizing equations (5.3.9) and (5.3.16) in equation (5.3.3) then at the boundary between
stress region #1 and stress region #2
106
[σYT 0 00 σYT 00 0 σYT
] A1 + [0 0 00 2σ YT 00 0 0 ]B1
+ [σYT 0 00 2 σYT 00 0 σ YT
] E1 + [0 0 00 2 σYT 00 0 0 ] F1
¿ [σ YT 0 00 σYT 00 0 σYT
] A2 + [0 0 00 2 σYT 00 0 0 ]B2
+ [0 0 00 0 00 0 σYT
]C2 + [σYT 0 00 2 σYT 00 0 σYT
] E2 + [0 0 00 2 σYT 00 0 0 ] F2
(5.3.17)
The following three expressions can be extracted from equation (5.3.19), i.e.,
A1 + E1 = A2 + E2 (5.3.18)
A1 + 2B1 + 2 E1 + 2 F1 = A2 + 2 B2 + 2 E2 + 2F2 (5.3.19)
and
A1 + E1 = A2 + C2 + E2 (5.3.20)
From equations (5.3.18) and (5.3.20) it is apparent that
C2 = 0(5.3.21)
From equations (5.3.19) and (5.3.18) it can be shown that
2 B1 + E1 + 2 F1 = 2B2 + E2 + 2 F2 (5.3.22)
For the stress state in region #3 given above the unit principle stress vector is
107
3 ai = (0 , 1 , 0 )(5.3.23)
Thus
(3 ai ) (3 a j) = [0 0 00 1 00 0 0 ]
(5.3.24)
The stress invariants associated with this unit vector and stress state are
3 I 4 = σYT (5.3.25)
3 I5 = σYT 2
(5.3.26)
3 I8 = σYT (5.3.27)
and
3 I 9 = σYT2
(5.3.28)
With these stress invariants and stress state the gradient along the boundary for stress
region #3 is
108
∂ g3
∂ σ ij= −[σYT 0 0
0 σYT 00 0 σYT
] A3 − [0 0 00 2 σYT 00 0 0 ]B3
− [σYT 0 00 2 σYT 00 0 σ YT
]C3 − [σYT 0 00 2 σYT 00 0 σYT
] E3
− [0 0 00 2σ YT 00 0 0 ]F3 − [σYT 0 0
0 2 σ YT 00 0 σYT
]G3
− [0 0 00 2 σYT 00 0 0 ]H3
(5.3.29)
Utilizing equations (5.3.16) and (5.3.29) in equation (5.3.4) then at the boundary between
stress region #2 and stress region #3
[σYT 0 00 σ YT 00 0 σYT
] A2 + [0 0 00 2 σYT 00 0 0 ] B2
+ [0 0 00 0 00 0 σYT
]C2 + [σYT 0 00 2 σYT 00 0 σYT
] E2 + [0 0 00 2 σYT 00 0 0 ] F2
¿ [σYT 0 00 σ YT 00 0 σYT
] A3 + [0 0 00 2 σYT 00 0 0 ] B3
+ [σYT 0 00 2 σYT 00 0 σ YT
]C3 + [0 0 00 2 σYT 00 0 0 ] D3 + [σYT 0 0
0 2 σYT 00 0 σ YT
]E3
+ [0 0 00 2 σYT 00 0 0 ] F3 + [σYT 0 0
0 2σ YT 00 0 σYT
]G3 + [0 0 00 2 σYT 00 0 0 ]H 3
(5.3.30)
109
1x
2x
3x
idYC
YC
The following three expressions can be extracted from equation (5.3.30), i.e.,
A2 + E2 = A3 + C3 + E3 + G3 (5.3.31)
A2 + 2B2 + 2 E2 + 2 F2¿ A3 + 2 B3 + 2 C3 + 2 D3
+ 2 E3 + 2 F3 + 2G3 + 2H 3 (5.3.32)
and
A2 + C2 + E2 = A3 + C3 + E3 + G3 (5.3.33)
Subtracting equation (5.3.33) from equation (5.3.31) leads to
C2 = 0(5.3.34)
which has already been established. Subtracting equation (5.3.31) from equation (5.3.32)
leads to
2 B2 + E2 + 2 F2¿ 2 B3 + C3 + 2 D3 + E3 + 2 F3 + G3 + 2 H3 (5.3.35)
Next consider the following stress state at failure under a compression load with
the same material direction di = (0, 1, 0) as above, i.e.,
σ ij = [0 0 00 σYC 00 0 0 ]
(5.3.36)
110
The subscript C denotes a compressive failure strength compression stress and it is noted
that this strength is algebraically less than zero. The principle stresses are obviously
( σ1 , σ2 , σ 3) = ( 0 , 0 , σYC )(5.3.37)
At the boundary of region #3 and region #4 the first principle stress is zero, i.e.,
Similarly at the boundary of region #2 and region #3 the second principle
stress is zero, i.e., . At both boundaries we impose the requirements that the
gradients match, i.e.,
∂ g2
∂σ ij=
∂ g3
∂ σ ij(5.3.38)
∂ g3
∂σ ij=
∂ g4
∂ σ ij(5.3.39)
providing a smooth transition from one principle stress region to another. Here the first,
second, sixth and seventh invariants of stress are
I 1 = σYC (5.3.40)
I 2 = σYC2
(5.3.41)
I 6 σYC (5.3.42)
and
111
I 7 = σYC2
(5.3.43)
These stress invariants are common for stress regions #2, #3, and #4 given this state of
stress.
For the stress state in region # 2 the unit principal vector is
2 ai = (0 , 1 , 0 )(5.3.44)
thus
(2 ai ) ( 2 a j) = [0 0 00 1 00 0 0 ]
(5.3.45)
The stress invariants associated with this unit vector and stress state are
2 I 4 = σYC (5.3.46)
2 I 5 = σYC2
(5.3.47)
2 I 8 = σ YC (5.3.48)
and
2 I 9 = σYC2
(5.3.49)
With these stress invariants and stress state, the gradient along the boundary for stress
region #2 is
112
∂ g2
∂ σ ij= −[σYC 0 0
0 σYC 00 0 σYC
] A2 − [0 0 00 2σ YC 00 0 0 ]B2
− [σYC 0 00 2 σYC 00 0 σYC
]C2 − [0 0 00 2 σYC 00 0 0 ] D2
− [σYC 0 00 2 σYC 00 0 σYC
]E2 − [0 0 00 2 σYC 00 0 0 ]F2
− [σ YC 0 00 2σ YC 00 0 σ YC
]G2 − [0 0 00 2 σYC 00 0 0 ] H2
(5.3.50)
As noted earlier the unit vector associated with region #3 is
3 ai = (1 , 0 , 0 )(5.3.51)
thus
(3 ai ) ( 3 a j ) = [1 0 00 0 00 0 0 ]
(5.3.52)
The stress invariants associated with this unit vector and stress state are
3 I 4 = 0(5.3.53)
3 I5 = 0(5.3.54)
3 I8 = 0(5.3.55)
and
113
3 I 9 = 0(5.3.56)
With these stress invariants and stress state, the gradient along the boundary for stress
region #3 is
∂ g3
∂ σ ij= −[σYC 0 0
0 σYC 00 0 σ YC
] A3 − [0 0 00 2σYC 00 0 0 ]B3 − [σYC 0 0
0 0 00 0 0 ]C3
− [σ YC 0 00 2σ YC 00 0 σ YC
]E3 − [0 0 00 2 σYC 00 0 0 ] F3
(5.3.57)
Utilizing equations (5.3.50) and (5.3.57) in equation (5.3.38) then at the boundary
between stress region #2 and stress region #3
114
[σYC 0 00 σYC 00 0 σ YC
] A2 + [0 0 00 2 σYC 00 0 0 ]B2 + [σ YC 0 0
0 2 σYC 00 0 σ YC
]C2
+ [0 0 00 2 σ YC 00 0 0 ]D2 + [σ YC 0 0
0 2 σYC 00 0 σ YC
]E2 + [0 0 00 2 σYC 00 0 0 ]F2
+ [σYC 0 00 2 σYC 00 0 σYC
]G2 + [0 0 00 2 σYC 00 0 0 ]H 2
¿ [σYC 0 00 σYC 00 0 σYC
]A3 + [0 0 00 2 σYC 00 0 0 ]B3 + [σYC 0 0
0 0 00 0 0 ]C3
+ [σYC 0 00 2 σYC 00 0 σYC
] E3 + [0 0 00 2 σ YC 00 0 0 ]F3
(5.3.58)
The following three expressions can be extracted from equation (5.3.58), i.e.,
A2 + C2 + E2 + G2 = A3 + C3 + E3 (5.3.59)
A2 + 2 B2 + 2C2 + 2 D2
+ 2 E2 + 2F2 + 2G2 + 2 H 2¿ A3 + 2 B3 + 2 E3 + 2F3 (5.3.60)
and
A2 + C2 + E2 + G2 = A3 + E3 (5.3.61)
Subtracting equation (5.3.61) from equation (5.3.59) leads to
C3 = 0(5.3.62)
115
Subtracting equation (5.3.61) from equation (5.3.60) leads to
2 B2 + C2 + 2 D2 + E2 + 2 F2 + G2 + 2 H 2¿ 2B3 + E3 + 2 F3 (5.3.63)
With the invariants established in equations (5.3.40) through (5.3.43), the following
gradient for region #3 takes the following form
∂ g4
∂ σ ij= [σ YC 0 0
0 σ YC 00 0 σYC
] A4 + [0 0 00 2 σYC 00 0 0 ]B4
+ [σYC 0 00 2 σYC 00 0 σYC
]E4 + [0 0 00 2 σYC 00 0 0 ] F4
(5.3.64)
Utilizing equations (5.3.57) and (5.3.64) in equation (5.3.39) then at the boundary
between principle stress region #3 and principle stress region #4
[σYC 0 00 σYC 00 0 σ YC
]A3 + [0 0 00 2 σYC 00 0 0 ]B3
+ [σYC 0 00 0 00 0 0 ]C3 + [σYC 0 0
0 2 σYC 00 0 σYC
]E3 + [0 0 00 2 σYC 00 0 0 ] F3
¿ [σYC 0 00 σYC 00 0 σYC
]A4 + [0 0 00 2σ YC 00 0 0 ] B4
+ [σYC 0 00 2 σYC 00 0 σYC
]E4 + [0 0 00 2 σYC 00 0 0 ]F4
(5.3.65)
The following three expressions can be extracted from equation (5.3.65), i.e.,
116
1x
2x
3x
id
TT
TT
A3 + C3+ E3 = A4 + E4 (5.3.66)
A3 + 2 B3 + 2 E3 + 2F3 = A4 + 2 B4 + 2 E4 + 2 F4 (5.3.67)
and
A3 + E3 = A4 + E4 (5.3.68)
From equations (5.3.66) and (5.3.68) it can be shown that
C3 = 0(5.3.69)
which was established before. The following expression can be obtained from equations
(5.3.67) and (5.3.68)
2B3 + E3 + 2 F3 = 2 B4 + E4 + 2 F4 (5.3.70)
Next consider the following stress state at failure under a tensile load with the
same preferred material direction di = (0, 1, 0), i.e.,
σ ij = [σTT 0 00 0 00 0 0 ]
(5.3.71)
117
The first subscript T denotes as tress in the direction transverse to the strong direction of
the material and the second subscript T means tension (TT. The principle stresses
are
(σ1 , σ2 , σ 3) = (σTT , 0 , 0 )(5.3.72)
In order to satisfy the definitions given earlier for the principle stress regions at the
shared boundary of region #2 and region #3 the second principle stress must be zero, i.e.,
At the shared boundary between region #1 and region #2 the third principle
stress must be zero, i.e., . The stress state given above satisfies these stress
conditions, i.e., both and At both boundaries we impose the
requirements that the gradients match, i.e.,
∂ g1
∂σ ij=
∂ g2
∂ σ ij(5.3.73)
∂ g2
∂σ ij=
∂ g3
∂ σ ij(5.3.74)
providing a smooth transition from one principle stress region to another. Using the
stress state given above the first, second, sixth and seventh invariants are
I 1 = σTT (5.3.75)
I 2 = σTT 2
(5.3.76)
118
I 6 = 0(5.3.77)
and
I 7 = 0(5.3.78)
These four invariants are common to stress regions #1, #2 and #3 for the stress state
given above. With these stress invariants the following gradient can be formulated for
this stress state
∂ g1
∂ σ ij= −[σTT 0 0
0 σTT 00 0 σTT
] A1
− [2 σTT 0 00 0 00 0 0 ] B1 − [0 0 0
0 σ TT 00 0 0 ] E1
(5.3.79)
For the stress state in region # 2 the unit principal vector is
2 ai = (0 , 0 , 1 )(5.3.80)
Thus
(2 ai ) ( 2 a j ) = [0 0 00 0 00 0 1 ]
(5.3.81)
The invariants associated with this principle stress vector and stress state are
2 I 4 = 0(5.3.82)
119
2 I 5 = 0(5.3.83)
2 I 8 = 0(5.3.84)
and
2 I 9 = 0(5.3.85)
With these invariants and stress state the gradient along the boundary for stress region #2
is
∂ g2
∂ σ ij= −[σTT 0 0
0 σTT 00 0 σTT
] A2 − [2 σ TT 0 00 0 00 0 0 ]B2
− [0 0 00 0 00 0 σTT
]C2 − [0 0 00 σTT 00 0 0 ] E2
(5.3.86)
Utilizing equations (5.79) and (5.3.86) in equation (5.3.73) then at the boundary between
principle stress region #1 and principle stress region #2
[σTT 0 00 σTT 00 0 σTT
]A1 + [2σTT 0 00 0 00 0 0 ]B1 + [0 0 0
0 σTT 00 0 0 ] E1
¿ [σTT 0 00 σ TT 00 0 σTT
]A2 + [2 σTT 0 00 0 00 0 0 ]B2 + [0 0 0
0 0 00 0 σ TT
]C2
+ [0 0 00 σTT 00 0 0 ]E2
(5.3.87)
120
The following three relationships between functional constants can be extracted from
equation (5.3.87)
A1 + 2B1 = A2 + 2B2 + 2C2 (5.3.88)
A1 + E1 = A2 + E2 (5.3.89)
and
A1 = A2 + C2 (5.3.90)
Subtracting equation (5.3.90) from equation (5.3.88) yields
2B1 = 2 B2 + C2 (5.3.91)
Subtracting equation (5.3.90) from equation (5.3.89) leads to
E1 = E2 − C2 (5.3.92)
For the stress state stipulated above the unit principal vector in region # 3 is
3 ai = (1 , 0 , 0 )(5.3.93)
thus
(3 ai ) ( 3 a j) = [1 0 00 0 00 0 0 ]
(5.3.94)
And the stress invariants for this principle stress vector and stress state are
121
3 I 4 = (1 ) (1 ) (σ TT ) = σTT(5.3.95)
3 I5 = (1 ) (1 ) (σTT ) (σTT ) = σTT 2
(5.3.96)
3 I8 = (1 ) ak dk (0 )σ TT = 0(5.3.97)
and
3 I 9 = (1 ) al dl (0 ) σTT σTT = 0(5.3.98)
With these invariants and stress state the gradient along the boundary for stress region #3
is
∂ g3
∂ σ ij= −[σTT 0 0
0 σTT 00 0 σTT
] A3 − [2σ TT 0 00 0 00 0 0 ]B3
− [2 σTT 0 00 σ TT 00 0 σTT
]C3 − [2σ TT 0 00 0 00 0 0 ] D3 − [0 0 0
0 σTT 00 0 0 ]E3
(5.3.99)
Utilizing equations (5.86) and (5.3.99) in equation (5.3.74) then at the boundary between
principle stress region #2 and principle stress region #3
122
[σTT 0 00 σTT 00 0 σTT
]A2 + [2 σTT 0 00 0 00 0 0 ] B2
+ [0 0 00 0 00 0 σTT
]C2 + [0 0 00 σTT 00 0 0 ] E2
¿ [σTT 0 00 σ TT 00 0 σTT
] A3 + [2 σTT 0 00 0 00 0 0 ] B3
+ [2 σTT 0 00 σTT 00 0 σTT
]C3 + [2 σTT 0 00 0 00 0 0 ] D3 + [0 0 0
0 σTT 00 0 0 ] E3
(5.3.100)
The following three expressions can be extracted from equation (5.3.105)
A2 + 2B2 = A3 + 2 B3 + 2 C3 + 2 D3 (5.3.101)
A2 + E2 = A3 + C3 + E3 (5.3.102)
and
A2 + C2 = A3 + C3 (5.3.103)
Subtracting equation (5.3.102) from equation (5.3.101) leads to
2 B2 − C2 = 2B3 + C3 + 2 D3 (5.3.104)
Subtracting equation (5.3.103) from equation (5.3.102) leads to
E2 − C2 = E3 (5.3.105)
123
1x
2x
3x
id
TC
TC
Next consider the following stress state at failure under a compressive load with
the preferred material direction di = (0, 1, 0), i.e.,
σ ij = [σTC 0 00 0 00 0 0 ]
(5.3.106)
The first subscript T denotes a stress in the direction transverse to the strong direction of
the material, and the second subscript C denotes compression (TC. The principle
stresses are
(σ1 , σ2 , σ 3) = (0 , 0 , σTC)(5.3.107)
In order to satisfy the definitions given earlier for the principle stress regions at the
shared boundary of region #3 and region #4 the first principle stress must be zero, i.e.,
At the shared boundary between region #2 and region #3 the second principle
stress must be zero, i.e., . The stress state given above satisfies these stress
conditions, i.e., both and At both boundaries we impose the
requirements that the gradients match, i.e.,
∂ g2
∂σ ij=
∂ g3
∂ σ ij(5.3.108)
124
∂ g3
∂σ ij=
∂ g4
∂ σ ij(5.3.109)
providing a smooth transition from one principle stress region to another. Using the
stress state given above the first, second, sixth and seventh invariants are
I 1 = σTC (5.3.110)
I 2 = σTC2
(5.3.111)
I 6 = 0(5.3.112)
and
I 7 = 0(5.3.113)
These four invariants are common to stress regions #2, #3 and #4 for the stress state
given above. With these stress invariants the following gradient can be formulated for
this stress state
∂ g4
∂ σ ij= −[σTC 0 0
0 σTC 00 0 σ TC
] A4
− [2σ TC 0 00 0 00 0 0 ]B4 − [0 0 0
0 σ TC 00 0 0 ] E4
(5.3.114)
125
For the stress state in region # 2 the unit principal vector is
2 ai = (1, 0 , 0 )(5.3.115)
Thus
(2 ai ) ( 2 a j ) = [1 0 00 0 00 0 0 ]
(5.3.116)
The stress invariants for this principle stress vector are
2 I 4 = σTC (5.3.117)
2 I 5 = σTC2
(5.3.118)
2 I 8 = 0(5.3.119)
and
2 I 9 = 0(5.3.120)
With these stress invariants the following gradient can be formulated for this stress state
∂ g2
∂ σ ij= −[σTC 0 0
0 σTC 00 0 σ TC
] A2 − [2σ TC 0 00 0 00 0 0 ] B2
− [2σTC 0 00 σTC 00 0 σTC
]C2 − [2σTC 0 00 0 00 0 0 ] D2 − [0 0 0
0 σTC 00 0 0 ]E2
(5.3.121)
For the stress state in region #3 the unit principal vector is
126
3 ai = (0 , 1 , 0 )(5.3.122)
Thus
(3 ai ) (3 a j ) = [0 0 00 1 00 0 0 ]
(5.3.123)
And the stress invariants for this principle stress vector are
3 I 4 = 0(5.3.124)
3 I5 = 0(5.3.125)
3 I8 = 0(5.3.126)
and
3 I 9 = 0(5.3.127)
With these stress invariants the following gradient can be formulated for this stress state
∂ g3
∂ σ ij= −[σTC 0 0
0 σTC 00 0 σ TC
] A3 − [2σ TC 0 00 0 00 0 0 ]B3
− [0 0 00 σTC 00 0 0 ]C3 − [0 0 0
0 σTC 00 0 0 ] E3
(5.3.128)
Utilizing equations (5.128) and (5.3.135) in equation (5.3.115) then at the boundary
between principle stress region #2 and principle stress region #3
127
[σTC 0 00 σTC 00 0 σ TC
] A2 + [2 σTC 0 00 0 00 0 0 ]B2
+ [2 σTC 0 00 σTC 00 0 σTC
]C2 + [2 σTC 0 00 0 00 0 0 ] D2 + [0 0 0
0 σTC 00 0 0 ]E2
¿ [σTC 0 00 σTC 00 0 σTC
] A3 + [2 σTC 0 00 0 00 0 0 ]B3
+ [0 0 00 σ TC 00 0 0 ]C3 + [0 0 0
0 σTC 00 0 0 ] E3
(5.3.129)
The following three expressions can be extracted from equation (5.3.136), i.e.,
A2 + 2B2 + 2C2 + 2D2 = A3 + 2 B3 (5.3.130)
A2 + C2 + E2 = A3 + C3 + E3 (5.3.131)
and
A2 + C2 = A3 (5.3.132)
Subtracting equation (5.3.139) from equation (5.3.137) yields
2 B2 + C2 + 2D2 = 2 B3 (5.3.133)
Subtracting equation (5.3.139) from equation (5.3.138) yields
E2 = C3 + E3 (5.3.134)
128
Utilizing equations (5.121) and (5.3.135) in equation (5.3.116) then at the
boundary between principle stress region #3 and principle stress region #4
[σTC 0 00 σTC 00 0 σ TC
] A3 + [2 σTC 0 00 0 00 0 0 ]B3
+ [0 0 00 σ TC 00 0 0 ]C3 + [0 0 0
0 σTC 00 0 0 ] E3
¿ [σTC 0 00 σTC 00 0 σTC
] A4
+ [2 σTC 0 00 0 00 0 0 ] B4 + [0 0 0
0 σ TC 00 0 0 ]E4
(5.3.135)
The following three expressions can be extracted from equation (5.3.142)
A3 + 2 B3 = A4 + 2 B4 (5.3.136)
A3 + C3 + E3 = A4 + E4 (5.3.137)
A3 = A4 (5.3.138)
Subtracting equation (5.3.145) from equation (5.3.143) leads to
B3 = B4 (5.3.139)
Subtraction equation (5.3.145) from equation (5.3.144) leads to
C3 + E3 = E4 (5.3.140)
129
Equations (5.3.23), (5.3.24), (5.3.37), (5.3.66), (5.3.67), (5.3.73), (5.3.96),
(5.3.97), (5.3.109), (5.3.110), (5.3.140), (5.3.141), (5.3.146) and (5.3.147) represent
fourteen equations in terms of twenty unknowns. From these fourteen equations the
following additional relationships between the coefficients are obtained
A1 = A2 = A3 = A4 (5.3.148)
B1 = B2 = B3 + D3 = B4 + D3 (5.3.149)
C2 = C3 = 0(5.3.150)
D2 = −D3 (5.3.151)
E1 = E2 = E3 = E4 (5.3.152)
F1 = F2 = F3 + H3 = F4 + H3 (5.3.153)
G2 = G3 = 0(5.3.154)
and
H2 = −H 3 (5.3.155)
130
1x
2x
3x
YT
YT
id
These relationships represent requirements that insure the four functional forms for the
failure function are smooth and continuous along the boundaries of the four stress
regions.
5.4 Functional Constants in Terms of Strength Parameters
Next we utilize specific load paths in order to define the constants defined above
in terms of strength values obtained in mechanical failure tests. Consider the following
stress state at failure under a uniaxial tensile load in the preferred material direction di =
(0, 1, 0)
σ ij = [0 0 00 σYT 00 0 0 ]
(5.4.1)
The principle stresses for this state of stress are
(σ1 , σ2 , σ 3) = (σYT , 0 , 0 )(5.4.2)
Note that one principle stress is tensile and the others are zero. This state of stress lies
along the border of region #1, region #2 and region #3. For region #1 the first, second,
sixth and seventh invariants are
I 1 = σYT (5.4.3)
131
I 2 = σYT 2
(5.4.4)
I 6 = σYT (5.4.5)
and
I 7 = σYT 2
(5.4.6)
With the failure function defined as
g1 = 1 − [(12 ) A1 I
12 + B 1 I 2 + E 1 I1 I 6 + F 1 I7 ]¿ 0 (5.4.7)
in region #1 of the principle stress space, then substitution of equations (5.4.3) through
(5.4.6) into the (5.4.7) yields
0 = 1 − [( 12 ) A1 (σYT )2 + B 1 σ
YT2 + E 1 σ YT σYT + F 1 σYT2 ]
(5.4.8)
or
( 12 ) A1 + B 1 + E 1 + F 1 = 1
σYT2
(5.4.9)
The unit principal stress vector associated with this state of stress in the region # 2
is
2 ai = (0 , 0 , 1 )(5.4.10)
thus
132
(2 ai ) ( 2 a j) = [0 0 00 0 00 0 1 ]
(5.4.11)
The stress invariants associated with this principle stress vector and state of stress are
2 I 4 = 0(5.4.12)
2 I 5 = 0(5.4.13)
2 I 8 = 0(5.4.14)
and
2 I 9 = 0(5.4.15)
The failure function for stress region #2 has the form
g2 = 1 − [ (1/2 ) A2 I12 + B 2 I 2 + C2 I1 I 4 + D2 I 5
+ E2 I1 I6 + F2 I7 + G2 I 1 I8 + H2 I 9 ]¿ 0 (5.4.16)
With equations (5.4.3) through (5.4.6) as well as equations (5.4.12) through (5.4.15) then
0 = 1 − [( 12 ) A2 (σYT )2 + B 2σ
YT2 + E2 (σYT ) (σ YT ) + F2 σYT2]
(5.4.17)
or
133
( 12 ) A2 + B 2 + E2 + F2 = 1
σYT2
(5.4.18)
The unit principal stress vector associated with this state of stress in the region # 3
is
3 ai = (0 , 1 , 0 )(5.4.19)
thus
(3 ai ) (3 a j) = [0 0 00 1 00 0 0 ]
(5.4.20)
The stress invariants associated with this principle stress vector and state of stress are
3 I 4 = σYT(5.4.21)
3 I5 = σYT 2
(5.4.22)
3 I8 = σYT (5.4.23)
and
3 I 9 = σYT2
(5.4.24)
The failure function for stress region #3 has the form
g3 = 1 − [ (1/2 ) A3 I12 + B 3 I 2 + C3 I 1 I 4 + D3 I5 I 7
+ E3 I 1 I6 + F3 + G3 I1 I 8 + H 3 I 9 ]¿ 0 (5.4.25)
134
1x
2x
3x
idYC
YC
With equations (5.4.3) through (5.4.6) as well as equations (5.4.21) through (5.4.24) then
0 = 1 − [(12 ) A3 (σYT )2 + B 3 σYT2 + D3 σ
YT2
+ E3 ( σYT ) (σYT ) + F3 σYT 2 + H3 σ
YT2 ](5.4.26)
and
( 12 ) A3 + B 3 + D3 + E3 + F3 + H3 = 1
σYT 2
(5.4.27)
Next consider the following stress state at failure due to a compressive stress
aligned with the material direction di = (0, 1, 0), i.e.,
σ ij = [0 0 00 σYC 00 0 0 ]
(5.4.28)
The principle stresses are
( σ1 , σ2 , σ 3) = ( 0 , 0 , σYC )(5.4.29)
Note that one principle stress is compressive and the others are zero. This state of stress
lies along the border of region #2, region #3 and region #4. For region #4 the first,
second, sixth and seventh invariants for this stress state are
I 1 = σYC (5.4.30)
135
I 2 = σYC2
(5.4.31)
I 6 = σYC (5.4.32)
and
I 7 = σYC2
(5.4.33)
With the failure function defined as
g4 = 1 − [ (1/2 ) A4 I12 + B4 I 2 + E4 I 1 I6 + F4 I 7]
¿ 0 (5.4.34)
in region #4 of the principle stress space, then substitution of equations (5.4.30) through
(5.4.33) into the (5.4.34) yields
0 = 1 − [(12 ) A4 (σYC )2 + B 4 σ
YC2 + E 4 (σ YC ) (σYC ) + F 4 σYC2]
(5.4.35)
or
( 12 ) A4 + B 4 + E 4 + F4 = 1
σYC2
(5.4.36)
The unit principal stress vector associated with this state of stress in region #2 is
2 ai = (0 , 1 , 0 )(5.4.37)
thus
136
(2 ai ) ( 2 a j) = [0 0 00 1 00 0 0 ]
(5.4.38)
The stress invariants associated with this principle stress vector and state of stress are
2 I 4 = σYC (5.4.39)
2 I 5 = σYC2
(5.4.40)
2 I 8 = σ YC (5.4.41)
and
2 I 9 = σYC2
(5.4.42)
Again, the failure function for stress region #2 has the form
g2 = 1 − [ (1/2 ) A2 I12 + B 2 I 2 + C2 I1 I4 + D2 I 5
+ E2 I 1 I 6 + F2 I7 + G2 I 1 I8 + H2 I9 ]¿ 0 (5.4.43)
With equations (5.4.30) through (5.4.33) as well as equations (5.4.39) through (5.4.42)
then
0 = 1 − [(12 ) A2 (σYC )2 + B 2 σYC2 + D2 σ
YC2
+ E2 ( σYC ) (σ YC ) + F2 σYC2 + H 2 σ
YC2 ](5.4.44)
or
137
( 12 ) A2 + B2 + D2 + E2 + F2 + H2 = 1
σYC2
(5.4.45)
The unit principal stress vector associated with this state of stress in region #3 is
3 ai = (1 , 0 , 0 )(5.4.46)
thus
(3 ai ) ( 3 a j ) = [1 0 00 0 00 0 0 ]
(5.4.47)
The stress invariants associated with this principle stress vector and state of stress are
3 I 4 = 0(5.4.48)
3 I5 = 0(5.4.49)
3 I8 = 0(5.4.50)
and
3 I 9 = 0(5.4.51)
Again, the failure function for stress region #3 has the form
g3 = 1 − [ (1/2 ) A3 I12 + B 3 I 2 + C3 I 1 I 4 + D3 I5 I 7
+ E3 I 1 I6 + F3 + G3 I1 I 8 + H 3 I 9 ]¿ 0 (5.4.52)
138
1x
2x
3x
id
TT
TT
With equations (5.4.30) through (5.4.33) as well as equations (5.4.46) through (5.4.49)
then
0 = 1 − [( 12 ) A3 ( σYC )2 + B 3 σ
YC2 + E3 (σYC ) ( σYC ) + F3 σYC2]
(5.4.53)
or
( 12 ) A3 + B 3 + E3 + F3 = 1
σYC2
(5.4.54)
Next consider the following stress state at failure due to a tensile stress
perpendicular to the material direction di = (0, 1, 0), i.e.,
σ ij = [σTT 0 00 0 00 0 0 ]
(5.4.55)
The principle stresses are
(σ1 , σ2 , σ 3) = (σTT , 0 , 0 )(5.4.56)
Note that principle stress aligned transverse to the preferred direction of the material is
tensile and the others are zero. This state of stress lies along the border of region #1,
139
region #2 and region #3. For region #1 the first, second, sixth and seventh invariants are
obtained
I 1 = σTT (5.4.57)
I 2 = σTT 2
(5.4.58)
I 6 = 0(5.4.59)
and
I 7 = 0(5.4.60)
Again, with the failure function defined as
g1 = 1 − [(12 ) A1 I
12 + B 1 I 2 + E 1 I1 I 6 + F 1 I7 ]¿ 0 (5.4.61)
in region #1 of the principle stress space, then substitution of equations (5.4.57) through
(5.4.60) into the (5.4.61) leads to
0 = 1 − [( 12 ) A1 (σTT )2 + B 1 σ
TT2 ](5.4.62)
or
( 12 ) A1 + B 1 = 1
σTT 2
(5.4.63)
The unit principal stress vector associated with this state of stress in region #2 is
140
2 ai = (0 , 0 , 1 )(5.4.64)
thus
(2 ai ) ( 2 a j ) = [0 0 00 0 00 0 1 ]
(5.4.65)
The stress invariants associated with this principle stress vector and state of stress are
2 I 4 = 0(5.4.66)
2 I 5 = 0(5.4.67)
2 I8 = 0(5.4.68)
and
2 I 9 = 0(5.4.69)
Again, the failure function for stress region #2 has the form
g2 = 1 − [ (1/2 ) A2 I12 + B 2 I 2 + C2 I1 I4 + D2 I 5
+ E2 I 1 I 6 + F2 I7 + G2 I 1 I8 + H2 I9 ]¿ 0 (5.4.70)
With equations (5.4.57) through (5.4.60) as well as equations (5.4.66) through (5.4.69)
then
0 = 1 − [( 12 ) A2 ( σTT )2 + B 2 σ
TT2 ](5.4.71)
141
or
( 12 ) A2 + B 2 = 1
σTT2
(5.4.72)
The unit principal stress vector associated with this state of stress in region #3 is
3 ai = (1 , 0 , 0 )(5.4.73)
thus
(3 ai ) ( 3 a j) = [1 0 00 0 00 0 0 ]
(5.4.74)
The stress invariants associated with this principle stress vector and state of stress are
3 I 4 = σTT (5.4.75)
3 I5 = σTT 2
(5.4.76)
3 I8 = 0(5.4.77)
and
3 I 9 = 0(5.4.78)
Again, the failure function for stress region #3 has the form
g3 = 1 − [ (1/2 ) A3 I12 + B 3 I 2 + C3 I 1 I 4 + D3 I5 I 7
+ E3 I 1 I6 + F3 + G3 I1 I 8 + H 3 I 9 ]¿ 0 (5.4.79)
142
1x
2x
3x
id
TC
TC
With equations (5.4.57) through (5.4.60) as well as equations (5.4.75) through (5.4.78)
then
0 = 1 − [( 12 ) A3 (σTT )2 + B 3σ
TT2 + D3 σTT2]
(5.4.80)
or
( 12 ) A3 + B 3 + D 3 = 1
σTT2
(5.4.81)
Next consider the following stress state at failure due to a compressive stress
perpendicular to the material direction di = (0, 1, 0), i.e.,
σ ij = [σTC 0 00 0 00 0 0 ]
(5.4.82)
The principle stresses are
(σ1 , σ2 , σ 3) = (0 , 0 , σTC)(5.4.83)
Note that one principle stress is compressive and the others are zero. This state of stress
lies along the border of region #2, region #3 and region #4. For region #4 the first,
second, sixth and seventh invariants are
I 1 = σTC (5.4.84)
143
I 2 = σTC2
(5.4.85)
I 6 = 0(5.4.86)
and
I 7 = 0(5.4.87)
Again, with the failure function defined as
g4 = 1 − [ (1/2 ) A4 I12 + B4 I 2 + E4 I 1 I6 + F4 I 7]
¿ 0 (5.4.88)
in region #4 of the principle stress space, then substitution of equations (5.4.84) through
(5.4.87) into the (5.4.88) leads to
0 = 1 − [( 12 ) A4 (σ TC )2 + B 4 σ
TC2](5.4.89)
or
( 12 ) A4 + B 4 = 1
σTC2
(5.4.90)
The unit principal stress vector associated with this state of stress in region #2 is
2 ai = (1, 0 , 0 )(5.4.91)
thus
144
(2 ai ) ( 2 a j ) = [1 0 00 0 00 0 0 ]
(5.4.92)
The stress invariants associated with this principle stress vector and state of stress are
2 I 4 = σTC (5.4.93)
2 I 5 = σTC2
(5.4.94)
2 I 8 = 0(5.4.95)
and
2 I 9 = 0(5.4.96)
Again, the failure function for stress region #2 has the form
g2 = 1 − [ (1/2 ) A2 I12 + B 2 I 2 + C2 I1 I4 + D2 I 5
+ E2 I 1 I 6 + F2 I7 + G2 I 1 I8 + H2 I9 ]¿ 0 (5.4.97)
With equations (5.4.84) through (5.4.87) as well as equations (5.4.93) through (5.4.96)
then
g2 = 1 − [( 12 ) A2 (σTC )2 + B 2 σ
TC2 + D2 σTC2 ]
(5.4.98)
or
145
( 12 ) A2 + B 2 + D2 = 1
σTC2
(5.4.99)
The unit principal stress vector associated with this state of stress in region #3 is
3 ai = (0 , 1 , 0 )(5.4.100)
thus
(3 ai ) (3 a j ) = [0 0 00 1 00 0 0 ]
(5.4.101)
The stress invariants associated with this principle stress vector and state of stress are
3 I 4 = 0(5.4.102)
3 I5 = 0(5.4.103)
3 I8 = 0(5.4.104)
and
3 I 9 = 0(5.4.105)
Again, the failure function for stress region #3 has the form
g3 = 1 − [ (1/2 ) A3 I12 + B 3 I 2 + C3 I 1 I 4 + D3 I5 I 7
+ E3 I 1 I6 + F3 + G3 I1 I 8 + H 3 I 9 ]¿ 0 (5.4.106)
146
1x
2x
3x
MBC
id
MBC
MBC
MBC
With equations (5.4.84) through (5.4.87) as well as equations (5.4.102) through (5.4.105)
then
0 = 1 − [( 12 ) A3 (σTT )2 + B 3σ
TT2 + D3 σTT2]
(5.4.107)
or
( 12 ) A3 + B 3 = 1
σTC2
(5.4.108)
Next consider the following stress state at failure due to an equal biaxial
compressive stress where one component of the applied stress is directed along the
material direction di = (0, 1, 0), i.e.,
σ ij = [σ MBC 0 00 σ MBC 00 0 0 ]
(5.4.109)
The subscript “MBC” denotes “mixed equal-biaxial-compression” and because the
applied stress is compressive algebraically then C. The principle stresses are
( σ1 , σ2 , σ 3) = ( 0 , σ MBC , σ MBC )(5.4.110)
147
Note that two principle stresses are compressive and the other is zero. This state of stress
lies along the border region #3 and region #4. For stress region #4 the first, second, sixth
and seventh invariants are
I 1 = 2 σ MBC (5.4.111)
I 2 = 2σMBC 2
(5.4.112)
I 6 = σ MBC (5.4.113)
and
I 7 = σMBC 2
(5.4.114)
Again, with the failure function defined as
g4 = 1 − [ (1/2 ) A4 I12 + B4 I 2 + E4 I 1 I6 + F4 I 7]
¿ 0 (5.4.115)
in region #4 of the principle stress space, then substitution of equations (5.4.111) through
(5.4.114) into the (5.4.115) leads to
0 = 1 − [(12 ) A4 (2σ MBC )2 + B 4 (2σMBC2)
+ E 4 (2 σMBC )(σ MBC ) + F 4 σMBC2 ]
(5.4.116)
or
2 A4 + 2B 4 + 2E 4 + F4 = 1σ
MBC2(5.4.116)
148
The unit principal stress vector associated with this state of stress in region #3 is
3 ai = (0 , 0 , 1 )(5.4.117)
thus
(3 ai ) (3 a j ) = [0 0 00 0 00 0 1 ]
(5.4.118)
The stress invariants associated with this principle stress vector and state of stress are
3 I 4 = 0(5.4.119)
3 I5 = 0(5.4.120)
3 I8 = 0(5.4.121)
and
3 I 9 = 0(5.4.122)
The failure function for stress region #3 now has the form
g3 = 1 − [ (1/2 ) A3 I12 + B 3 I 2 + C3 I 1 I 4 + D3 I5 I 7
+ E3 I 1 I6 + F3 + G3 I1 I 8 + H 3 I 9 ]¿ 0 (5.4.123)
With equations (5.4.111) through (5.4.114) as well as equations (5.4.119) through
(5.4.122) then
149
1x
2x
3x
BC
BC
id
BC
BC
0 = 1 − [(12 ) A3 (2 σ MBC )2 + B 3 (2 σMBC2)
+ E3 (2 σ MBC ) (σ MBC ) + F3σMBC2 ]
(5.4.124)
or
2 A3 + 2 B 3 + 2 E3 + F3 = 1σ
MBC2(5.4.125)
Next consider the following stress state at failure under a biaxial equal
compression load
σ ij = [σBC 0 00 0 00 0 σ BC
](5.4.126)
The subscript “BC” denotes “biaxial-compression” and because the stress is compressive
then algebraically C. Also note that the stresses are applied in the plane of
isotropy. The principle stresses are
(σ1 , σ2 , σ 3) = (σ BC , 0 , σ BC )(5.4.127)
This state of stress lies along the border between region #3 and region #4. For this state
of stress the first, second , sixth and seventh invariants are obtained
I 1 = 2 σ BC (5.4.128)
150
I 2 = 2 σBC 2
(5.4.129)
I 6 = 0(5.4.130)
and
I 7 = 0(5.4.131)
Again, with the failure function defined as
g4 = 1 − [ (1/2 ) A4 I12 + B4 I 2 + E4 I 1 I6 + F4 I 7]
¿ 0 (5.4.132)
in region #4 of the principle stress space, then substitution of equations (5.4.127) through
(5.4.130) into the (5.4.131) leads to
0 = ( 12 ) A4 ( 2σBC )2 + B 4 ( 2σ
BC 2) − 1
(5.4.133)
or
2 A4 + 2 B 4 = 1σ
BC2(5.4.134)
The unit principal stress vector associated with this state of stress in region #3 is
3 ai = (0 , 1 , 0 )(5.4.135)
thus
151
(3 ai ) (3 a j ) = [0 0 00 1 00 0 0 ]
(5.4.136)
The stress invariants associated with this principle stress vector and state of stress are
3 I 4 = 0(5.4.137)
3 I5 = 0(5.4.138)
3 I8 = 0(5.4.139)
and
3 I 9 = 0(5.4.140)
The failure function for stress region #3 now has the form
g3 = 1 − [ (1/2 ) A3 I12 + B 3 I 2 + C3 I 1 I 4 + D3 I5 I 7
+ E3 I 1 I6 + F3 + G3 I1 I 8 + H 3 I 9 ]¿ 0 (5.4.141)
With equations (5.4.128) through (5.4.131) as well as equations (5.4.136) through
(5.4.139) then
0 = 1 − [( 12 ) A3 ( 2σBC )2 + B 3 (2σ
BC2)](5.4.142)
or
152
2 A3 + 2 B 3 = 1σ
BC2(5.4.143)
(5.4.97) - (5.4.57)
B4 − B 1 = 1σ
TC2− 1
σTT2
= D2
(5.4.130)
(5.4.49) - (5.4.9)
( B 4 − B 1 ) + ( F 4 − F 1 ) = 1σ
YC2− 1
σYT2
(5.4.131)
(F 4 − F 1) = ( 1σ
YC2− 1
σYT2 ) − ( 1
σTC2
− 1σ
TT 2) = H2
(5.4.132)
(5.4.121) – 2 * (5.4.97)
A4 = 1σ
BC2− 2
σTC2
(5.4.133)
Substitution of equation (5.4.133) into (5.4.97)
B 4 = 2σ
TC2− 1
2σBC2
(5.4.134)
Substitution of equation (5.4.133) into (5.4.57)
153
B 1 = 1σ
TT2− 1
2 σBC2
+ 1σ
TC2(5.4.135)
(5.4.105) - (5.4.49)
32
A4 + B 4 + E 4 = 1σ
MBC2− 1
σYC2
(5.4.136)
Substitution of equations (5.4.133) and (5.4.134) into (5.4.136)
E 4 = 1σ
MBC2− 1
σYC2
− 1σ
BC2+ 1
σTC2
(5.4.137)
(5.4.105) - (5.4.121)
2 E 4 + F4 = 1σ
MBC2− 1
σBC2
(5.4.138)
Substitution of equation (5.4.137) into (5.4.138)
F4 = − 1σ
MBC2+ 1
σBC 2
+ 2σ
YC2− 2
σTC2
(5.4.139)
Substitution of equation (5.4.139) into (5.4.132)
F 1 = − 1σ
MBC2+ 1
σBC 2
+ 1σ
YC2− 1
σTC2
+ 1σ
YT2− 1
σTT2
(5.4.140)
So the coefficients are
A1 = A2 = A3 = A4 = − 2σ
TC2+ 1
σBC 2
(5.4.141)
154
B1 = B2 = B3 + D3 = B4 + D3
= 1σ
TT2+ 1
σTC2
− 12 σ
BC 2(5.4.142)
D2 = − D3 = − 1σ
TT2+ 1
σTC2
(5.4.143)
E1 = E2 = E3 = E4
¿ 1σ
TC2− 1
σBC2
− 1σ
YC2+ 1
σMBC2
(5.4.144)
F1 = F2 = F3 + H3 = F4 + H3
¿ − 1σ
TT2− 1
σTC2
+ 1σ
BC2
+ 1σ
YT2+ 1
σYC2
− 1σ
MBC2 (5.4.145)
H2 = −H 3 = 1σ
TT2− 1
σTC2
− 1σ
YT2+ 1
σYC2
(5.4.146)
where the strength parameters identified above are defined as
YT – tensile strength in the preferred material direction
YC – compressive strength in the preferred material direction
TT – tensile strength in the plane of isotropy
TC – compressive strength in the plane of isotropy
BC – equal biaxial compressive strength in the plane of isotropy
MBC – equal biaxial compressive strength with only one stress component
in the plane of isotropy
155
The anisotropic Green and Mkrtichian (1977) failure criterion is projection onto
the 11 – 22 stress space in Figure 5.4.1. The strength parameters were for the most part
once again extracted from Burchell’s (2007) data., i.e., TT =10.48 MPa, YT =15.93 MPa
and YC =52.93 MPa (see Table 2.2). These are average or mean strength values. The
other three strength parameters YT, YC, BC and MBC were estimated. Values for the
strength parameters listed above are given in the figure caption. Note the agreement with
the data along the two tensile axes, as well as along the failure curve for each load path.
These average strength values for each load path are depicted as open red circles in
Figure 5.4.1.
156
MPa93.15,0
, 2211
MPaMPa 4.61,4.61
, 2211
MPa93.52,0
, 2211
1
1
)(11 MPa
)(22 MPa
0,48.10
, 2211
MPa
0,0.35
, 2211
MPa
Figure 5.4.1 Anisotropic Green-Mkrtichian (1977) failure criterion with Burchell’s (2007) failure data projected onto the 11 - 22 principle stress plane (TT = 10.48 MPa, TC = 35MPa, BC = 40 MPaYT = 15.93 MPa, YC = 52.93 MPa, MBC = 61.40 MPa)
The anisotropic Green and Mkrtichian (1977) failure criterion is projected onto
the deviatoric planes in Figures 5.4.2, 5.4.3 and 5.4.4 Note that a cross section through
the failure function perpendicular to the hydrostatic axis transitions from a pyramidal
shape (Figure 5.4.1) to a circular shape (Figure 5.4.2) with an increasing value of the
stress invariant I1. This suggests that the apex of the failure function presented in a full
Haigh-Westergaard stress space is blunt, i.e., quite rounded for the this particular
criterion.
157
11
2
3 2
3
O0
O60O120
O180
O270
O300
MPa10
MPa30
MPa50
oMPaMPar 0,56.8,05.6,,
Figure 5.4.2 Anisotropic Green-Mkrtichian (1977) failure criterion with Burchell’s (2007) failure data projected onto (= 6.05 MPa) parallel to the -plane(TT = 10.48
MPa, TC = 35MPa, BC = 40 MPaYT = 15.93 MPa, YC = 52.93 MPa, MBC = 61.40 MPa)
158
11
2
3 2
3
O0
O60O120
O180
O270
O300
MPa10
MPa30
MPa50
oMPaMPar 120,01.13,20.9,,
Figure 5.4.3 Anisotropic Green-Mkrtichian (1977) failure criterion with Burchell’s (2007) failure data projected onto (= 9.02 MPa) parallel to the -plane(TT = 10.48
MPa, TC = 35MPa, BC = 40 MPaYT = 15.93 MPa, YC = 52.93 MPa, MBC = 61.40 MPa)
159
160
11
2
3 2
3
O0
O60O120
O180
O270 O300
MPa10
MPa30
MPa50
oMPaMPar 300,17.42,2.30,,
Figure 5.4.4 Anisotropic Green-Mkrtichian (1977) failure criterion with Burchell’s (2007) failure data projected onto (= -30.20) parallel to the -plane(TT = 10.48 MPa,
TC = 35MPa, BC = 40 MPaYT = 15.93 MPa, YC = 52.93 MPa, MBC = 61.40 MPa)
The meridian lines of the anisotropic Green-Mkrtichian (1977) failure surface
corresponding to θ=0oθ=120o
andθ=300o are depicted on Figure 5.4.5 Obviously
the meridian lines are not linear. The θ=0omeridian line goes through point defined
161
oMPaMPar 300,17.42,2.30,,
oMPaMPar 0,56.8,05.6,,
oMPaMPar 0,01.813,20.9,,
oMPaMPar 0,13.50,9.70,,
MPa
MPar
by 6.05MPa and r = 8.56 MPa. θ=120omeridian line goes through point
defined by 9.02 MPa and r = 13.01 MPa. The θ=300omeridian line goes through
the point defined by = 30.2 MPa and r = 42.17MPa.
Figure 5.4.5 Anisotropic Green-Mkrtichian (1977) failure criterion with Burchell’s (2007) failure data projected onto meridian-plane(TT = 10.48 MPa, TC = 35MPa, BC =
40 MPaYT = 15.93 MPa, YC = 52.93 MPa, MBC = 61.40 MPa)
162
As the value of the I1 stress invariant associated with the hydrostatic stress
increases in the negative direction, failure surfaces perpendicular to the hydrostatic stress
line become circular again. The model suggests that as hydrostatic compression stress
increases the difference between tensile strength and compressive strength diminishes
and approach each other asymptotically. This is a material behavior that should be
verified experimentally in a manner similar to Bridgman’s (1953) bend bar experiments
conducted in hyperbaric chambers on cast metal alloys. Balzer (1998) provides an
excellent overview of Bridgman’s experimental efforts, as well as others and their
accomplishments in the field of high pressure testing.
163
CHAPTER VI
MONTE CARLO METHODS USING IMPORTANCE SAMPLING
A failure function defines a limit state through the design parameters the function
is dependent on. In this effort all design parameters are related to strength, although one
can easily pose limit states for fracture (e.g., failure assessment diagrams), fatigue life or
service issues relating to structural deformations. Since the design variables here are
restricted to strength parameters, the strength parameters associated with the each failure
function discussed previously in this dissertation can be assembled into an n-dimensional
vector, i.e.,
Y α = (Y 1 ,Y 2 ,⋯,Y n)
and the limit state function is in simple terms
g ( yα ) = 0
164
In simple terms this last expression defines a surface in an n-dimensional stress space if
the strength parameters are interpreted as stress values, which has been done throughout.
So the last expression should appear as
g ( yα , σ ij ) = 0
for clarity. If the stress state lies within the surface then this represents a safe operational
state at that point in the component. If the stress state lies on the surface then this
constitutes a failed operational state. This concept easily merges with interpreting failure
through the use of factors of safety. Inside the surface would correspond to a factor of
safety greater than one. On the surface, or outside the surface corresponds to a factor of
safety less than one.
The fundamental issue of no longer treating material strength as a deterministic,
single valued parameter complicates the issue of how to interpret failure at a point. If
strength is treated as a random variable, how does that affect the approach the design
engineer takes in assessing whether a component works properly or not? A different
design philosophy should be adopted in this situation where a simple fail/no fail
interpretation is replaced with an equivalent stochastic approach that predicts the
probability of component failure instead of the mainstream factor of safety calculations.
One should recognize that all parameters in an engineering design can be treated
as random variables, e.g., loads applied to the component as well as the stiffness of the
165
material utilized. However, the assumption is made that the variability in material
strength far exceeds the variability one would see in the other design parameters. This
seems to be a valid assumption since the strength of graphite material can vary by 50% or
more. The exception to this assumption would be designs where wind loads and/or
earthquake loads must be considered, although including load design parameters as well
resistance design parameters as random variables can be easily accomplished.
Several other assumptions are made in computing the probability of failure (or
reliability) of a component using the limit state functions presented here. First, it is
assumed that strength parameters utilized represent independent random variables. That
assumption was not interrogated here, but methods to approach this issue will be outlined
in the summary chapter. Another assumption is made relative to the probability density
function used to represent the strength random variables. Here the two parameter
Weibull distribution is adopted for all random strength parameters. Again there are
methods to test the validity of that assumption. But since this is a proof of concept effort,
those sort of goodness of fit tests are left to others to pursue.
In general, the reliability (probability of failure) is computed based on the
expression
R = Probability [ g ( yα , σ ij ) < 0 ]
166
This calculation is made for a unit volume of material that maintains a homogenous state
of stress. To calculate the reliability of an element the joint density function must be
integrated over design space defined by the limit state function. This integration takes
the form
R = ∭g ( yα , σ ij) < 0
Ω ( yT , yC , y BC) dyT dyC dyBC
(6.1)
for isotropic formulation of the Green-Mkrtichian limit state function where the three
random strength parameters Yt, Yc and Ybc were defined in the previous chapter. Note that
lower case letters associated with the random strength parameters are realizations of the
random strengths. In addition
R = ∫g ( yα , σ ij) < 0
Ω ( yTT , yTC , yBC , yYT , yYC , y MBC ) dyTT dyTC dy BC dyYT dyYC dyMBC
(6.2)
for anisotropic formulation of the Green-Mkrtichian limit state function.. Here Ωis the
joint density function of material strength parameters. Each strength random variable is
characterized by a two parameter Weibull distribution.
As Sun and Yamada (1978) as well as Wetherhold(1983) point out, the
integration defined by equations (6.1) and (6.2) yields the reliability of a unit volume.
introduced a technique to calculate the integration. Closed form solutions for these two
expressions are not available. However, integration using Monte Carlo simulation is
readily available.
167
6.1 Monte Carlo Simulation - Concept
In general the probability of failure of a structural component can be expressed as
Pf = ∫δ f
f Y α( yα ) dy
(6.1.1)
Here Y is a random strength variable, fY is a probability density function associated with
the random strength variable and δf is the failure domain that satisfies the expression
g ( yα ) ≤ 0 (6.1.2)
where g(y) is the functional representation of the failure criterion. Although the integral
seems straight forward, closed form solutions are unavailable except for simple failure
criterion. As an alternative, conventional Monte Carlo simulation is commonly used to
numerically evaluate the probability of failure when a closed form solution is difficult to
formulate.
Monte Carlo simulation is relatively easy to implement. An indicator function I is
defined such that
I = {1 g ( yα ) ≤ 00 g ( yα ) > 0
(6.1.3)
This indicator function can be included in the integral above if the integration range is
expanded to include the entire design variable space. Thus
Pf = ∫δ f +δ f
I f ( y α ) dyα
(6.1.4)
168
where earlier δs was defined as the safe domain of the design variable space. The integral
on the right side of this expression defines the expectation of the indicator function, i.e.,
E [ I ] = ∫δ f +δ f
I f ( y α )dyα
(6.1.5)
Recall from statistics that the definition of the mean (μ) of a random variable (say X) is
the expectation of the variable. Thus
μx = ∫−∞
+∞x f ( x ) dx
(6.1.6)
Also recall that the mean associated with a random variable can be estimated from a
sample taken from the population that is being characterized by the distribution function
f(x). The estimated value of the mean is given by the simple expression
μx = 1N ∑
j=1
N
x j
(6.1.7)
Where xj is the jth observation in a random sample taken from the population. In a similar
fashion the probability of failure (Pf) represents the mean (or expected value) of the
indicator function. Thus equation (6.1.5) can be expressed as
Pf = limN →∞ { 1
N ∑j=1
N
x j} (6.1.8)
Here it is implied that a random sample of successes (I = 1) or failures (I = 0) has been
generated. Thus Ij is the jth evaluation of the limit state function where the random
observations have been generated from the cumulative distribution function FX.
169
6.2 The Concept of Importance Sampling Simulation
As noted above Monte Carlo simulation is computationally simple. To increase
the accuracy of this numerical integration method the number of samples is simply
increased. However, the method does not converge to correct answers in the low
probability of failure regime. As engineers we wish to design components with very low
probabilities of failure. Thus conventional Monte Carlo simulation is modified using
importance sampling to achieve this goal. With importance sampling the design space is
sampled only within the near vicinity of the most probable point (MPP – see Figure
3.1.2). The name for the technique is derived from the notion that more sampling should
take place in the most "important" region of the design variable space. The important
region of the design variable space corresponds to the MPP. Thus the method requires a
general knowledge of the location of the MPP. However, determining the exact location
of the MPP is not necessary – just the general vicinity of the MPP. We note this method
alleviates a potential non-conservative numerical error associated with the fast probability
integration method. The limit state function in this work is by no means a linear function
in terms of the design random variables and when using fast probability integration (FPI)
techniques the limit state function is approximated by a hyper-plane at the MPP. As
Wetherhold and Ucci (1994) point out that a planar approximation can yield non-
conservative results depending on the curvature of the limit state function at the MPP.
170
Since importance sampling does not depend on this curvature, it effectively avoids this
potential non-conservative numerical error.
For a low probability of failure the main contribution to Pf will come from
regions near the MPP. This region (see Figure 6.2.1) will also correspond to the tail of
the joint probability distribution function of the design strength random variables.
Harbitz (1986) has shown that restricting the sampling domain in the design variable
space to the tail of the joint probability distribution function produces a remarkable
increase in efficiency in comparison to conventional Monte Carlo techniques.
Figure 6.2.1 The principle of the importance samplingIn order to develop the details of importance sampling, express equation (4.2.5) in
the following manner
Pf = ∫δf +δ f
If Y α
( yα )kY α
( yα )kY α
( yα ) dyα
(6.2.1)
171
Here Y is a vector of random strength parameters. The function Yk serves as the
probability density function for an alternative indicator function defined as
I = If Y α
( yα )kY α
( yα )kY α
( yα ) (6.2.2)
Equation (6.2.1)can be expressed as a Riemann sum, i.e.,
Pf = limN →∞ {1N ∑
j=1
N
I j [ f Yα( yα )
kY α( yα ) ]}
= limN →∞ {1N ∑
j=1
N
I j} (6.2.3)
Thus a random number is generated and realizations for each of the random strength
variables (Y)j are computed using the inverse of YK , which is the cumulative
distribution function corresponding to Yk in the equation above. Both functions have yet
to be defined. These realizations are then used to determine values of Yk , Yf and the
limit state function g(y) (which yields a value of the indicator function I).
Harbitz (1986) demonstrated that the number of simulations necessary to achieve
the same order of accuracy for the conventional Monte Carlo methods is reduced by a
factor of
172
1/ { 1 − Γ α ( ( β¿)2 )} (6.2.4)
where Γα is the chi-square distribution with degrees of freedom, and * is less than or
equal to the actual the reliability index for a given problem. The degrees of freedom
correspond to the number of design variables included in the limit state function. In
essence, Harbitz (1986) reasoned that random design variables are being sampled from a
truncated distribution function. This corresponds to sampling from the actual probability
distribution function, however the sampling domain is restricted to regions outside a
sphere defined in the design variable space (see Figure 6.2.1). The center of the sphere is
located at the origin of the transformed design variable space, and the radius of the sphere
is equal to *. Proof of Harbitz (1986) argument follows from the interpretation of this
geometrical concept.
In practice Yk should be selected such that sampling of the random variable
takes place in a small region surrounding the MPP. The importance sampling concept is
illustrated in Figure 6.2.1 which depicts a two-dimensional transformed design space.
The rational is easily extended to an n-dimensional random variable space. As noted
above the key is sampling around the MPP. One readily applied procedure is to select
Yk such that the mean ( Y
) of this probability density function lies near the MPP. The
173
FPI method is utilized to obtain approximate Z* ( β) values to establish a general
location of the MPP. Keep in mind that for two random variables
β = [ (z1¿ )2 + ( z2¿)2]1/2
(6.2.5)
Three or more random variables would be a simple extension of this geometric concept.
Here zα is the vector of standard normal variables which are related to the design
variables in the following manner
zα =( yα − μYα )
δY α (6.2.6)
Here y are realizations of the random strength variable Y, and values correspond to
graphite strength parameters outlined in previous sections of the report.
With an approximate z* value the mean associated with the probability density
function Yk is given by the following expression
μk Yα
= z1¿ [δ f ( x )] + μ f ( x) (6.2.7)
The parameters μ f ( x ) and δ f ( x ) are the mean and standard deviation associated with the
actual probability distribution function that characterizes the random strength variable,
i.e., fX. The strength random variables are assumed characterized by a two parameter
Weibull distribution. The standard deviation of the probability density function Yk is
174
chosen in such a way that the sampling region is restricted to the near vicinity of the
MPP. Here an approach suggested by Melchers (1989) is adopted where
δ kY α
= (1→3 )( μkYα
μf ( x ))δf ( x )
(6.2.8)
Although this procedure does not limit the type of distribution for Yk a normal
distribution is assumed here for simplicity, i.e.,
kY α= kY α
( yα )
= (1δ kY α
√2 π ) exp[−12 ( yα − μkYα
δkY α
)2
] (6.2.9)
This is a standard normal probability density function.
Although the transformed random strength variable has a standard normal
formulation, the parent distributionYf
is still implicitly embedded in the computations
above. Every potential parent distribution has a mean and standard deviation. At this
point the focus of the discussion turns to the assumption that the random strength
variables are characterized by a two parameter Weibull distribution.
175
6.3 Two Parameter Weibull Distribution
The strength of graphite material is essentially stochastic. Therefore the
stochastic method must be used to a probability of failure graphite for a component
instead of using factors of safety. Weibull(1939) formulated a fracture probability with a
two-parameter function under simple tension. The two parameters Weibull distribution
become well known and are widely used. The reliability of an element of unit volume for
uniaxial stress is
R = exp[−( σβ )
α ] (6.4.1)
where is the applied stress, is Weibull modulus and is scale parameter.
This equation is linearized by taking the natural logarithm of both sides of the
expression twice, then
ln (ln( 1R )) = ln (C ) + α ln ( σ )
(6.4.2)
where
ln (C ) = ( 1β )
α
(6.4.3)
The direct method of calculation of failure probability is integral function. As
discussion above, it is impossible when the failure function is complex especially in the
176
real world problem. Therefore, the First Order Reliability Method(FORM) and Second
Order Reliability Method(SORM) are often used in probability design. Taylor expansion
is used to optimize failure function. Both methods reliability index, corresponding to
MPP, accuracy is not satisfied. The Importance Sampling Simulation based on the
MPP(or design points) known previously is utilized to increase the accuracy of the failure
probability calculation.
A Direct Importance Sampling simulation is suggested by Melchers(1989). In
this method a normal distribution function is considered as the sampling density function.
The results are shown in Figures 4.4.3 and 4.4.4. The accuracy of the curves is lower
than expectation. Obviously the results are not good enough to describe the behavior of
graphite component. Another disadvantage of the approach is very low efficiency. The
Figure 4.4.3 took about 83 hours only for one volume unit. It is not acceptable. Exploring
a high efficient and more accuracy is very necessary.
New sampling density function will be applied and comparison of the results.
177
ln
fP11lnln
ln
fP11lnln
Figure 6.4.1 Reliability estimates of uniaxial tensile strengths using 100 Importance Sampling Simulations
178
ln
fP11lnln
Figure 6.4.2 Reliability estimates of uniaxial tensile strengths using 1000 Importance Sampling Simulations
Figure 6.4.3 Reliability estimates of uniaxial tensile strengths using 10000 Importance Sampling Simulations
179
ln
fP11lnln
ln
fP11lnln
Figure 6.4.4 Reliability estimates of uniaxial compression strengths using 100 Importance Sampling Simulations
180
ln
fP11lnln
ln
fP11lnln
Figure 6.4.5 Reliability estimates of uniaxial compression strengths using 1000 Importance Sampling Simulations
Figure 6.4.6 Reliability estimates of uniaxial biaxial compression strengths using 100 Importance Sampling Simulations
181
Figure 6.4.7 Reliability estimates of uniaxial biaxial compression strengths using 1000 Importance Sampling Simulations
The figures about points ( from figure 4.4.3 to figure 4.4.10)show that the values
of importance sampling simulation is good in (Pf = 5% ) and (Pf = 50% ) . But it trends to
small in (Pf = 95% ) compared with the true value. Figures shows that simulation is better
in tension than in compression.
Assuming that each strength parameter is characterized by a two parameter
Weibull distribution, and with knowledge of the Weibull distribution parameters and
m for each strength parameter, then the expression
μ f ( x ) = σθ Γ (1+ 1m )
(6.4.4)
can be used to compute the mean for each random strength variable ( is the gamma
function) and
δ f ( x ) = (σθ )2 {Γ (1+ 2m ) − [ Γ (1+ 1
m )]2}
(6.4.5)
is used to calculate the standard deviation for the strength random variables YT ,YC and
YBC.
182
6.4 Application to the Green-Mkrtichian Limit State Functions
Since most structural design analyses will involve more than one random design
variable, an n-dimensional normal probability density must be utilized. The sampling
function is constructed assuming independent random variables, i.e.,
k ( y α ) = ∏α=1
n
kY α( y α )
(6.3.1)
For the isotropic formulation for the Green-Mkrtichian (1977) failure function n is three.
For the anisotropic formulation n is six.
The concept of importance sampling is first applied to the isotropic form of the
Green and Mkrtichian (1977) limit state function. The tensile strength design variable
(YT), compressive strength design variable (YC) and the biaxial compressive strength
design variable (YBC) are characterized by the two-parameter Weibull distributions. Here
kYT takes the following form
kY T( yT ) = ( 1
δ YT√2π ) exp[−1
2 ( yT − μYT
δ YT)
2] (6.3.2)
Similarly kYc take the following form
kY C( yC ) = ( 1
δYC√2π ) exp [−1
2 ( yC − μY C
δ YC)
2] (6.3.3)
and
183
kY BC( yBC ) = ( 1
δY BC√2 π ) exp[−1
2 ( yBC − μY BC
δY BC)
2] (6.3.4)
Thus the composite sampling function is given by the expression for the isotropic Green
and Mkrtichian (1977) limit state function
k j (Y T , Y C , Y BC ) = [kYT( yT )] j [ kY C
( yC ) ] j [kY BC( y BC ) ] j (6.3.5)
The joint probability density function for random strength has the following form
f j (Y T , Y C , Y BC ) = [ f Y T( yT ) ] j [ f Y C
( yC )] j [ f Y BC( y BC )] j (6.3.6)
In both functions it is assumed that YT ,YC and YBC are independent random variables.
Thus importance sampling for the Green and Mkrtichian (1977) limit state
function begins with approximate values for the transformed variables ZYT, ZYC and ZYBC.
These approximate values are obtained using the FPI technique. With these values of the
transformed variables then μYT, μYC and μYBC are obtained using equation (6.2.10). This
requires knowledge of the mean and the standard deviation of the strength distribution.
Thus δYT, δYC and δYBC are computed using equation (6.3.2). With μ and δ defined
for all three strength random variables, the probability density functions kYT , kYc and kYBC
can be formulated. In addition, the cumulative distribution functions KYT, K YC and KYBC are
defined. Next, a random number associated with each random variable (YT, YC and YBC) is
generated. These random numbers generate realizations of each random by utilizing the
inverses of KYT, K YC and KYBC. With realizations of the random variables, call them (yT)j,
184
(yC)j and (yBC)j, specific functional values for kYT , kYc and kYBC can be generated. Now a
functional value for kj(yT, yC , yBC ) can be generated using equation (6.2.6). In a similar
fashion a functional value for fj(yT, yC,, yBC) can be generated using equation (6.2.7).
Finally, the limit state function is computed using equations (4.2.2), (4.2.10), (4.2.24) as
well as (4.2.36) and this allows the computation of the indicator function using equation
(6.2.2). The quantities kj(yT, yC , yBC ), fj(yT, yC,, yBC) and I are inserted into equation (6.2.3)
and the summation is performed for a sufficient number of iterations (i.e., large enough
N) such that the method converges to Pf.
6.5 Contours of Equal Reliability
Projections of reliability surfaces are presented in Figure 3.1.2 for the isotropic
formulation of the Green and Mkrtichian (1977) limit state function outlined in a
previous section. Figure 3.1.3 depicts the reliability surfaces for the anisotropic version
of that same limit state function. Monte Carlo simulations with importance sampling
technique were utilized to generate the surfaces in both figures. The Weibull parameters
(m and σθ) for each design variable are
mT = 6.58 mC = 12.29 mBC = 13.99
T = 17.05 MPa C = 54.39 MPa BC = 63.29 MPa
for isotropy. For anisotropy the Weibull distribution parameters for tensile strength in the
preferred material direction are
185
mYT = 6.58
YT = 17.05 MPa
The Weibull distribution parameters for compressive strength in the preferred material
direction
mYC = 12.19
YC = 54.39 MPa
The Weibull distribution parameters for tensile strength in the plane of isotropy
mTT = 10.12
TT = 11.01 MPa
The Weibull distribution parameters for compressive strength in the plane of isotropy
mTC = 10.33
TC = 35.90 MPa
The Weibull distribution parameters for equal biaxial compressive strength in the plane
of isotropy
mBC = 11.85
BC = 45.95 MPa
The Weibull distribution parameters for equal biaxial compressive strength with only one
stress component in the plane of isotropy
mMBC = 13.99
MBC = 63.29 MPa
186
)(11 MPa
)(22 MPa
Three reliability surfaces are depicted in both figures that correspond to probabilities of
failure of Pf = 5% , Pf = 50% , and Pf = 95%.
Figure 6.5.1 Burchell’s (2007) failure data with probability of failure curves obtained using Monte Carlo simulation modified with importance sampling techniques for the
isotropic version of the Green Mkrtichian (1977) failure criterion.
187
)(11 MPa
)(22 MPa
Figure 6.5.2 Burchell’s (2007) failure data with probability of failure curves obtained using Monte Carlo simulation modified with importance sampling techniques for the
anisotropic version of the Green Mkrtichian (1977) failure criterion.6.6 ASME Reliability Curves
188
189
CHAPTER VII
SUMMARY AND CONCLUSIONS
Graphite is a brittle or quasi-brittle material characterized by the Weibull
distribution. Different material behavior in tension and in compression and material
anisotropy make formulating a failure model challenging. This work develops an
anisotropic formulation of Green-Mkrtichian failure criterion model based on the
invariant theory and a technique to predict life of graphite components applied in the
researching Generation IV nuclear power plants.
Two continuum and two piecewise continuous failure criterion models are
introduced. Von Mises failure criterion surface is a circular cylinder along with the
hydrostatic line. Its failure function is based on only one invariant I1. Drucker-Prager
failure criterion is described as a circular cone with the tip of cone located in positive –
axis. Its integrity base is constructed with invariants I1, J2. When the ratio of compressive
and tensile strength is greater than or equate to 3 the curve is open along with equal
biaxial compression load path. Willam-Warnke failure criterion is a pyramid shape with a
190
triangular base, which is piecewise continuous with a threefold symmetry. It is a function
of invariants I1, J2 and J3. Each segment is assumed as a portion of an approximate ellipse
on the deviatoric plane. The Lode angle is determined by invariants J2 and J3. Unlike
linear meridians of Von Mises and Drucker-Prager failure criterion the meridians vary in
different Willam-Warnke failure surface. Another piecewise model introduced is Green-
Mkrtichian failure criterion. The Green-Mkrtichian failure criterion considers the
principle stress unit vector a1 when both tensile and compressive stresses are loaded. The
invariant integrity base is invariants I1, I2, I4 and I5. The curve portioned Haigh-
Westergaard stress space and offered four forms for the failure functions depending on
different stress status. Compared with the complexity of calculation of radial vector r in
Willam-Warnke failure function, the Green-Mkrtichian failure function is quadratic
formula. It makes the isotropic invariant integrity base to be easily extended to the
anisotropic base.
The Burchell’s data shows that the graphite has slight anisotropic behavior in
different direction. The material direction must be considered. An anisotropic invariant
integrity base is created with the material unit vector Di, including nine invariants Ii (i =
1, 2, 3…9). In anisotropic formulation of Green-Mkrtichian failure criterion, only the six
invariants I1, I2, I5, I6, I7 and I9 construct the integrity base. And six coefficients are
determined by six stress tests, including two tensile stress tests, two compressive stress
tests and two equal biaxial compressive stress tests.
191
The graphite is essentially stochastic random. The deterministic failure model is
necessarily transformed to probabilistic failure model through treating the strength
variables as random variables. In the anisotropic formulation of Green-Mkrtichian failure
criterion there are six strength parameters, TT, TC,BC,YT,YC,MBC. Each strength
parameter is characterized by two parameter Weibull distribution. Maximum Likelihood
method is used for Weibull distribution parameter estimation based on the nine sets of
Burchell’s data of graphite H-451 samples. The reliability can be obtained by the
integrity of the Weibull’s reliability function under the simple uniaxial strength. But the
complexity of Green-Mkrtchian failure function makes it tough to get a closed form.
Therefore, the Monte Carlo simulation is introduced.
For the high risky in nuclear power plants, low failure probability of graphite
components is evaluated. It means large quantity of samples is required if Monte Carlo
simulation used, which is a numerically expensive operation. So the Monte Carlo
simulation with Importance Sampling is applied. In this technique the main interest is the
vicinity area around the point of MPP during the evaluation. FPI method is used to
calculate the reliability index corresponding to the MPP. Because the Green-
Mkrtichian failure function is nonlinear and the graphite is characterized by Weibull
distribution, several methods are utilized for the procedure of calculation. Rackwitz-
Fiessler method transforms Weibull-variables to normal-variables. The Hasofer-Lind
Approximation approximates nonlinear failure surface to linear. The failure probability of
192
unit graphite volume is easily obtained with . The Monte Carlo simulation with
Importance Sampling does different quantity of trials around the MPP. The entire
simulation is programmed by MATLAB. The different curves with three failure
probability, 5%, 50% and 95, are plotted.
The following general conclusions are extracted:
Classical Von Mises failure criterion and Drucker-Prager failure criterion are not
good enough to describe the anisotropic behavior of graphite material.
Isotropic formulation of Willam-Warnke failure criterion is tough to expand to
anisotropic formulation because of the complexity of the radial vector r
calculation.
There are only six invariants I1, I2, I5, I6, I7 and I9 are in the integrity base for
anisotropic formulation of Green-Mkrtichian failure criterion. The anisotropic
invariant integrity base is easily expanded by the isotropic integrity base because
of Green-Mkrtichian failure function is quadratic.
The anisotropic formulation of Green-Mkrtichian failure criterion model describes
the property behavior of graphite very well.
The Monte Carlo simulation with Importance Sampling predicts the reliability of
graphite with low quantity samples.
In the future, the research efforts should focus on improving the calculation
effective and applying the model and simulation to different fields:
193
Current work is only for failure probability calculation of graphite unit volume. A
large scale size component and effect size will be considered.
The anisotropic formulation of Green-Mkrtichian failure criterion model will be
applied to composite material
The Monte Carlo simulation with Importance Sampling technique will be inserted
into algorithms reliability estimation such as COMSOL.
The computation of failure probabilities by Monte Carlo simulation with
Importance Sampling is a numerical expensive operation for large structural
models. Optimization algorithm is necessary.
194
REFERENCES
Batdorf, S.B. and Crose, J.G., “A statistical Theory for the Fracture of Brittle Structure Subjected to Nonuniform Polyaxial Stress,” ASME Journal of Applied Mechanics, Vol. 41, No. 2, pp. 459-464, 1974.
Boehler, J. P. and Sawczuk, A., “On Yielding of Oriented Solids,” Acta Mechanica, Vol. 27, pp. 185-204, 1977.
Boehler, J. P., Kirillov, A.A., and Onat, E.T., “On the Polynomial Invariants of the Elasticity Tensor,” Journal of Elasticity, Vol 34, pp. 97-110, 1994.
Boehler, J. P., “Yielding and Failure of Transversely Isotropic Solids, Applications of Tensor Functions in Solid Mechanics,” CISM Courses and Lectures No 292, Springer-Verlag, Berlin, pp. 3-140, 1987.
Bridgman, P.W., “The Effect of Pressure on the Tensile Properties of Several Metals and Other Materials,” Journal of Applied Physics, Vol. 24, No. 5, pp.560-570, 1953.
Burchell, T., Yahr, T. and Battiste , R., “Modeling the Multiaxial Strength of H-451 Nuclear Grade Graphite,” Carbon , Vol. 45, No. 13, pp. 2570-2583, 2007.
Cazacu, O., Cristescu, N.D., Shao, J.F., and Henry, J.P., “A New Anisotropic Failure Criterion for Transversely Isotropic Solids,” Mechanics Cohesive-Frictional Material, Vol. 3, pp. 89-103, 1998.
Cazacu, O., and Cristescu, N.D., “A Paraboloid Failure Surface For Transversely Isotropic Materials,” Mechanics of Materials, Vol. 31, No. 6, pp. 381-393, 1999.
Chen, W.F., Plasticity in Reinforced Concrete, McGraw-Hill, Inc., pp. 190-250, 1982.
Chen, W.F. and Han, D.J., Plasticity for Structural Engineers, Gau Lih Book Co. Ltd., Taipei, Taiwan, 1995.
Cooper, N.R., Margetson, J., and Humble, S., “Probability Of Failure Calculations And Confidence Brand Estimates For An Annular Brittle Disc Fracture Under Centrifugal Loading,” Journal of Strain Analysis for Engineering Design, Vol. 21, No. 3, 121-126, 1986.
195
Cooper, N. R., “Probabilistic Failure Prediction of Rocket Motor Components,” Ph.D. Dissertation, Royal Military College of Science, 1988.
Coulomb, C. A., “Essai sur une application des regles des maximis et minimis a quelquels problemesde statique relatifs, a la architecture,” Memoires de Mathematique et de Physique, presentes a l’Academie Royales Des Sciences, Vol. 7, pp. 343–387, 1776.
Drucker, D.C., and Prager, W., “Soil Mechanics And Plastic Analysis For Limit Design,” Quarterly of Applied Mathematics, Vol. 10, No. 2, pp. 157–165, 1952.
Duffy, S.F., “A Viscoplastic Constitutive Theory for Transversely Isotropic Metal Alloys,” Ph.D. Dissertation, University of Akron, 1987.
Duffy, S.F., Gyekenyesi, J.P., “Time Dependent Reliability Model Incorporating Continuum Damage Mechanics for High-Temperature Ceramics,” NASA Technical Memorandum 102046, 1989.
Duffy, S.F., and Arnold, S.M., “Noninteractive Macroscopic Reliability Model for Whisker-Reinforced Ceramic Composites,” Journal of Composite Materials, Vol. 24, No. 3, pp.235-344, 1990a.
Duffy, S.F., and Manderscheid, J.M., "Noninteractive Macroscopic Reliability Model for Ceramic Matrix Composites with Orthotropic Material Symmetry," Journal of Engineering for Gas Turbines and Power, Vol. 112, No. 4, pp. 507 511, 1990b.
Duffy, S.F., Chulya, A. and Gyekenyesi, J.P., “Structural Design Methodologies for Ceramic-Based Material Systems,” NASA Technical Memorandum 103097, 1991.
Duffy, S.F., Wetherhold, R.C. and Jain, L.K., "Extension of a Noninteractive Reliability Mod for Ceramic Matrix Composites," Journal of Engineering for Gas Turbines and Power, Vol. 115, No. 1, pp. 205 207, 1993.
Duffy, S.F., and Palko, J.L., “Analysis of Whisker Toughened CMC Structural Components Using an Interactive Reliability Model,” AIAA Journal, Vol. 32, No. 5, pp. 1043-1048, 1994.
196
Duffy, S.F., Baker, E.H., and James, C., "Standard Practice for Reporting Uniaxial Strength Data And Estimating Weibull Distribution Parameters For Graphite," ASTM Standard D 7846 (Passed ASTM society ballot Fall 2012).
Green, A.E. and Mkrtichian, J.Z., “Elastic Solids with Different Moduli in Tension and Compression,” Journal of Elasticity, Vol. 7, No. 4, pp. 369-386, 1977.Griffiths, A. A., “The Theory Of Rupture And Flow In Solids,” Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol. 221, pp. 163-198, 1921.
Gyekenyesi, J.P., “SCARE- A Post-Processor Program to MSC/NASTRAN for the Reliability Analysis of Ceramic Components,” ASME Journal of Engineering for Gas Turbines and Power, Vol. 108, No. 3, 540-546, 1986.
Harbitz, A. “An Efficient Sampling Method for Probability of Failure Calculations,” Structural Safety, Vol. 3, pp. 108-115, 1986.
Hsieh, S.S., Ting, E.C., Chen, W.F., “An Elastic-Fracture Model for Concrete,” Proceedings 3rd Engineering Mechanics Division Specialists Conference, Austin, TX, pp. 437–441, 1979.
Lamon, J., “Reliability Analysis of Ceramics Using the CERAM Computer Program,” ASME Paper No. 90-GT-98, IGTI, 1990.
Li, Q.M., “Strain energy Density Failure Criterion,” International Journal of Solids and Structures, Vol. 38, No. 38, pp. 6997–7013, 2001.
Melcher, R.E., “Importance Sampling in Structural Systems, “ Structural Safety, Vol. 6, pp. 3-10, 1989
Nova, R., and Zaninetti, A., “An Investigation Into the Tensile Behavior of a Schistose Rock,” International Journal of Rock Mechanics, Mining Sciences and Geomechanics, Vol. 27, No. 4, pp. 231-242, 1990.
Palko, Joseph L., “An Interactive reliability Model for Whisker- Toughened Ceramics,” Masters Thesis, June, 1992.
Paul, B., “Generalized Pyramidal Fracture and Yield Criteria,” International Journal of Solids and Structure, Vol. 4, pp.175-196, 1968.
197
Rivlin, R.S. and Smith, G.F., “Orthogonal Integrity Basis for N Symmetric Matrices,” in Contributions to Mechanics, ed. D. Abir, Pergamon Press, Oxford, pp. 121-141, 1969.
Saito, S., “Role of Nuclear Energy to a Future Society of Shortage of Energy Resources and Global Warming,” Journal of Nuclear Materials, Vol. 398, pp. 1-9, 2010.Schleicher, F., “Der Spannungszustand an der Fließgrenze (Plastizitätsbedingung),” Zeitschrift für Angewandte Mathematik und Mechanik, Vol. 6, pp. 199- 216, 1926.
Spencer, A.J.M., “Theory of Invariants,” in Continuum Physics, 1, ed. A.C. Eringen, Academic Press, London, pp. 259-353, 1971.
Spencer, A.J.M., “Continuum Theory of the Mechanics of Fibre-Reinforced Composites,” Spring-Verlag, pp.1-32, 1984.
Tabaddor, F., “A Survey of Constitutive Educations of Bimodulus Elastic Materials,” in Mechanics of Bimodulus Materials, ed: C.W. Bert, Proceedings of the American Society of Mechanical Engineering Winter Annual meeting, AMD33, 2–5 (1979).
Tresca, H., “Mémoire sur l'écoulement des corps solides soumis à de fortes pressions,” Comptes Rendus, Academie de Science, Vol. 59, p. 754, 1864.
Tsai, S.W. and Wu, E.M., “A General Theory of Strength of Anisotropic Materials,” Journal of Composite Materials, Vol. 5, pp. 58-80, 1971.Theocaris, P.S., “The Elliptic Paraboloid Failure Criterion for Cellular Solids and Brittle Foams,” Acta Mechanica, Vol. 89, pp. 93-121, 1991.
Wetherhold, R.C. and Ucci, A.M., “Probability Techniques for Reliability Analyses of Composite Materials,” NASA CR-195294, National Aeronautics and Space Administration, National Technical Information Service, 1994.
US DOE Nuclear Energy Research Advisory Committee and the Generation IV International Forum, “A Technology Roadmap for Generation IV Nuclear Energy Systems,” GIF-002-00, December, 2002.
Vijavakumar, K. and Ashoka, J.G., “A Bilinear Constitutive Model for Isotropic Bimodulus Materials,” ASME Journal of Engineering Materials and Technology, Vol. 112, pp. 372-379, 1990.
198
Vijayakumar, K., and Rao, K.P., “Stress-Strain Relations for Composites with Different Stiffness in Tension and Compression,” Computational Mechanics, Vol. 2, pp. 167-175, 1987.
Von Mises, R., “Mechanik der festen Korper im plastisch deformablen Zustand,” Göttinger Nachrichten, pp. 582–592, 1913.Weibull, W., “A Statistical Distribution Function of Wide Applicability,” Journal of Applied Mechanics, Vol. 18, No. 3, pp. 293-297, 1951.
Weibull, Waloddi. Ingeniors Vetenshaps Akadamein, Handlingar, 151(1939)
Willam, K.J. and Warnke, E.P., “Constitutive Models for the Triaxial Behavior of Concrete,” in: Proceedings of the International Association for Bridge and Structural Engineer Seminar on Structures Subjected to Triaxial Stresses (Bergamo, Italy), Vol. 19, pp. 1-31, 1974.
199
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